Wisconsin has a straw purchase law, but I don't think the criteria were met as Kyle Rittenhouse was not a prohibited person as defined under
941.29 (1m):
https://docs.legis.wisconsin.gov/statutes/statutes/941/iii/2905
941.2905 Straw purchasing of firearms.
(1) Whoever intentionally furnishes, purchases, or possesses a firearm for a person, knowing that the person is prohibited from possessing a firearm under s.
941.29 (1m), is guilty of a Class G felony.
(2) The prohibition in sub.
(1) against possessing a firearm for a person who is prohibited from possessing a firearm does not apply to the possession of a firearm by any of the following:
(a) A person to whom the firearm is surrendered under s.
813.1285.
(b) A person who has been designated under s.
51.20 (13) (cv) 3. to store the firearm during the duration of the order under s.
51.20 (13) (cv) 1. not to possess a firearm.
(c) A person who has been designated under s.
51.45 (13) (i) 3. to store the firearm during the duration of the order under s.
51.45 (13) (i) 1. not to possess a firearm.
(d) A person who has been designated under s.
54.10 (3) (f) 3. to store the firearm during the duration of the order under s.
54.10 (3) (f) 1.
(e) A person who has been designated under s.
55.12 (10) (c) to store the firearm during the duration of the order under s.
55.12 (10) (a).
(f) A person not covered under pars.
(a) to
(e) who has been designated to store the firearm during the duration of any temporary prohibition on the possession of a firearm.
Which leaves federal law. There is no federal 'straw purchase' law, but it is a felony to make a false statement on ATF form 4473.
Dominick Black has admitted under oath in open court that Kyle Rittenhouse gave him the money to purchase the M&P 15, and that Kyle picked out which firearm to purchase.
Question 21.a on the current
ATF 4473:
21.a: Are you the actual transferee/buyer of the firearm(s) listed on this form and any continuation sheet(s) (ATF Form 5300.9A)? Warning: You are not the actual transferee/buyer if you are acquiring the firearm(s) on behalf of another person. If you are not the actual transferee/buyer, the licensee cannot transfer the firearm(s) to you. Exception: If you are only picking up a repaired firearm(s) for another person, you are not required to answer 21.a. and may proceed to question 21.b.
...
I certify that my answers in Section B are true, correct, and complete. I have read and understand the Notices, Instructions, and Definitions on ATF Form 4473. I understand that answering “yes” to question 21.a. if I am not the actual transferee/buyer is a crime punishable as a felony under Federal law, and may also violate State and/or local law.
The instructions on ATF 4473 for question 21.a are:
Question 21.a. Actual Transferee/Buyer: For purposes of this form, a person is
the actual transferee/buyer if he/she is purchasing the firearm for him/herself or
otherwise acquiring the firearm for him/herself. (e.g., redeeming the firearm from
pawn, retrieving it from consignment, firearm raffle winner). A person is also the
actual transferee/buyer if he/she is legitimately purchasing the firearm as a bona
fide gift for a third party. A gift is not bona fide if another person offered or gave
the person completing this form money, service(s), or item(s) of value to acquire
the firearm for him/her, or if the other person is prohibited by law from receiving or
possessing the firearm.
EXAMPLES: Mr. Smith asks Mr. Jones to purchase a firearm for Mr. Smith (who
may or may not be prohibited). Mr. Smith gives Mr. Jones the money for the firearm.
Mr. Jones is NOT THE ACTUAL TRANSFEREE/BUYER of the firearm and must
answer “no” to question 21.a. The licensee may not transfer the firearm to Mr.
Jones. However, if Mr. Brown buys the firearm with his own money to give to Mr.
Black as a gift (with no service or tangible thing of value provided by Mr. Black),
Mr. Brown is the actual transferee/buyer of the firearm and should answer “yes” to
question 21.a. However, the transferor/seller may not transfer a firearm to any
person he/she knows or has reasonable cause to believe is prohibited under 18
U.S.C. 922(g), (h), (n), or (x).
So the question comes down to whether Kyle Rittenhouse needed to have practical possession of the firearm after the purchase in order for Dominick Black to have made a false statement to question 21.a of ATF form 4473. The rifle was stored at Dominick Black's step father's house, and Kyle had access to it on a couple of occasions, including the night of August 25, 2020. Does having access to the firearm constitute possession?
I cannot speculate whether Dominick Black's purchase was made "on behalf of another person."