Of course, scholars usually refrain from posting items such as these because they are worried their ideas will get stolen before they can publish them in an academic journal--but I trust you. . .
The connection between the sacrifices of Exodus 24 and atonement language in the Targumim appears to flow naturally from the logic of Leviticus.
To begin with, let us point out that the sacrifices in Exodus from which Moses takes the “blood of the covenant” are explicitly identified as “peace offerings” (שׁלם) (cf. Exod 24:5).
Now, in Leviticus 17 it is said that such sacrifices are to be taken to the Tabernacle. Israelites are not to offer them elsewhere. They must bring them to the Tabernacle for a specific purpose: so that the priest can sprinkle the blood from the victims at the door of the sanctuary. Notice that it is specifically "peace-offerings" (also known as "well-being" offerings) which are explicitly mentioned:
This is to the end that the people of Israel may bring their sacrifices which they slay in the open field, that they may bring them to the Lord, to the priest at the door of the tent of meeting, and slay them as sacrifices of peace offerings to the Lord; 6 and the priest shall sprinkle the blood on the altar of the Lord at the door of the tent of meeting, and burn the fat for a pleasing odor to the Lord. 7 So they shall no more slay their sacrifices for satyrs, after whom they play the harlot. This shall be a statute for ever to them throughout their generations (Lev 17:5–7).Leviticus 17 then goes on to explain that anyone who does not bring their burnt offerings or sacrifices to the Tabernacle will be cut off from Israel.
And you shall say to them, Any man of the house of Israel, or of the strangers that sojourn among them, who offers a burnt offering or sacrifice [זבח] and does not bring it to the door of the tent of meeting, to sacrifice it to the Lord; that man shall be cut off from his people (Lev 17:8–9).Two offerings are specifically mentioned now: the burnt offering and the zebah [זבח]. Has the peace-offering fallen out of the picture? I don't think so. Kurtz points out that within the Pentateuch the term zebah is exclusively used to describe the peace-offering. This suggests that the peace-offering is therefore still in view.
So why is God so insistent that sacrifices be brought to the Tent so that the blood from the victims can be sprinkled? Well, from the following verses it would seem that―at least in part―the command has to do with concern that Israelites not consume the blood of the offering.
If any man of the house of Israel or of the strangers that sojourn among them eats any blood, I will set my face against that person who eats blood, and will cut him off from among his people (Lev 17:10).With this we've at last reached the critical verse. Here the Lord explains the precise reason for the injunction against drinking blood:
For the life of the flesh is in the blood; and I have given it for you upon the altar to make atonement for your souls; for it is the blood that makes atonement, by reason of the life. 12 Therefore I have said to the people of Israel, No person among you shall eat blood, neither shall any stranger who sojourns among you eat blood” (Lev 17:11–12).Now here’s my question: given the context, which kind of sacrificial offering is in view here? I would submit that it is possible that since peace-offerings have been in view throughout the chapter the passage was read in connection with those specific sacrifices. Indeed, the concern would make sense: peace-offerings are eaten thus there you have the concern for the blood. So even though peace-offerings are not specifically attached to atonement elsewhere the readers of the Targums could make the connection between them.
In Exodus 24:5 then Israelites offer peace offerings at the covenant ceremony at Sinai. The authors of Targumim made the connection between them and atonement―a connection which could possibly be made in Leviticus 17. In fact, since in Exodus 24 it is the blood (“the blood of the covenant”) that is specifically in view and since in Leviticus 17 blood is specifically tied with atonement (“it is the blood that makes atonement”), the connection was easy to make.
The Last Supper and Atonement Language
How does this relate to the Last Supper? Well, notice that Jesus seems to speak of his blood having an atoning value. This is seen in multiple ways.
1. Jesus' language of his blood being "poured out"--something found in all three Synoptic accounts of the Last Supper evokes the Levitical law code. Not only does it evoke the language of Leviticus 17, but also the fact that the blood of the sacrificial animals brought for atonement had to be "poured out" (cf. Lev 4:7, 18, 25, 30, 34).
2. The ritual of pouring out blood is also linked with the Day of Atonement in the Dead Sea Scrolls (cf. 11Q19 15:3; 23:13). These texts provide strong support for the antiquity of the traditions found in the Mishna which also link the pouring out of blood to the Yom Kippur liturgy (cf. m. Yoma 5:4, 7; cf. also b. Yoma 56b). The Talmud explains that the blood was poured from cups (cf. b. Yoma 57b). This may be significant. There is a particularly striking parallel between Jesus’ words and Sirach 50:15, which explain that on the Day of Atonement the duties of the high priest apparently involved “pouring out” (ἐξέχεεν) the “blood of the grape” (αἵματος σταφυλῆς) from a “cup” (σπονδείου) (cf. Sir 50:15). That Jesus has spoken of his “blood” being “poured out” in connection with the wine in the “cup” is strikingly evocative of this text.
3. In Isaiah 53 we read about the Suffering Servant who “poured out his soul to death. . . he bore the sin of many” (Isa 53:12). That atonement imagery is linked to the Suffering Servant is clear. He is explicitly described as a “sin-offering,” who, like the scapegoat of Yom Kippur, is said to “bear iniquities" and "he bore the sin of many" (Isa 53:10, 12). Of course, it is widely accepted that Jesus' saying about his blood being "poured out for many" in Matthew and Mark (cf. Matt 26:27//Mark 14:24) is drawing on this prophecy.
4. In the account of the Last Supper in 1 Corinthians 11, Paul does not mention Jesus’ blood being “poured out.” But this does not mean that he does not see it as having atoning value. It is possible that the very image of Jesus' "blood" would have evoked such imagery for Paul. Expiation is typically associated with Jesus' blood throughout the New Testament books, including in other Pauline letters (cf. Rom 3:25; 5:9; Eph 1:7; 2:13; Col 1:20; Heb 9:12, 14; 10:19, 29, 12:24; 13:12; 1 Pet 1:2, 19; 1 John 1:7; 5:6, 8; Rev 1:5; 5:9; 7:14; 12:11). Confirmation that Paul has this in mind may be seen in the following.
5. That Jesus’ dies for others is explicitly stated in the Lukan version of the bread-saying: "This is my body which is given for you" (τοῦτό ἐστιν τὸ σῶμά μου τὸ ὑπὲρ ὑμῶν διδόμενον; Luke 22:19). While Paul simply has "for you" (ὑπὲρ ὑμῶν), omitting "given" (διδόμενον), most commentators rightly note that he probably intends the same meaning―Jesus is giving his life for others. Indeed, elsewhere Paul uses the preposition ὑπὲρ (="for") to describe Christ’s death as an expiatory sacrifice (e.g., 1 Cor 15:3; Rom 5:6, 8). An allusion to atonement imagery is thus likely present in his account.
6. Of course, Matthew specifically has Jesus' describing his blood being poured out "for the forgiveness of sins". Some have seen here a reference to Isaiah 53, others to Jeremiah 31, and still others think both are in mind.
7. It may be significant that Jesus describes himself not only as a sacrifice but also as an edible offering (i.e., he gives his “body” to be eaten). In this scholars we might have an allusion to the Passover sacrifice, which was a kind of peace-offering. That would be significant because Jesus' sacrifice would then be linked with the same kinds of sacrifice offered in Exodus 24--a passage clearly evoked by his words which link his "blood" to "covenant".
However, there is another kind of offering which might also be mentioned: the sin-offering. Leviticus 10:17 seems to suggest that the priest's eating of the sin-offering was intrisically linked to atonement. After Aaron and the priests fail to eat of the sacrifices, Moses states: "Why have you not eaten the sin offering in the place of the sanctuary, since it is a thing most holy and has been given to you that you may bear the iniquity of the congregation, to make atonement for them before the Lord?" (Lev 10:17). That eating of the sacrifice was an essential part of making atonement is thus recognized by many scholars of Israel's cultic laws [e.g., Milgrom (Leviticus 1-16, 638), Gane (Cult and Character, 96), and Levine (Leviticus, 62)]. By themselves imagery of “blood” being “poured out” and the eating of the sacrifice are merely possible points of contact with atonement terminology. However their appearance alongside each other within a passage containing allusions to the Suffering Servant (=a sin-offering) and the “covenant” ceremony of Exodus 24 makes an allusion to the eating of sin-offerings highly probable.
In closing. . .
Obviously, I realize that Jesus' language of "new covenant" may also have carried atonement implications--prophecies of God establishing/renewing an eschatological covenant with his people were almost always associated with the idea of reconciliation with God (i.e., the former covenant had been broken) so atonement imagery might have simply been suggested by that connection.
Either way, it seems atonement is in view in a passage in which Jesus cites from Exodus 24--a passage the Targums also linked with atonement.
Much more could be said. You'll have to read my thesis to get more.
But keep in mind, I’m just thinking out loud here.
 For the following see J. H. Kurtz, Sacrificial Worship of the Old Testament (trans. J. Martin; Edinburgh: T & T Clark, 1863; repr., Grand Rapids: Baker House,1980), 365.