205 ILCS 5/45.1) (from Ch. 17, par. 356)
Sec. 45.1. Accounts for minors. A state bank may accept deposits made by a minor and may open an account in the name of such minor and the rules and regulations of such bank with respect to each such deposit and account shall be as binding upon such minor as if such minor were of full age and legal capacity. The receipt, acquittance or order of payment of such minor on such account or deposit or any part thereof shall be as binding upon such minor as if such minor were of full age and legal capacity.
(Source: Laws 1965, p. 2020.)
Does this mean you can open an account for a minor in IL without a custodian?

Yup.

Ok, another question - does Illinois require the custodian to turn the funds over to the beneficiary/minor at 18??

21 or 18, depending on the type of transfer. See section 21 of 760 ILCS 20.

Note that the law you cited allows you to open an account in the name of a minor. It says nothing about joint accounts. I suggest you get an opinion from counsel on whether you can legally open a joint account with a minor as one of the joint owners.