With no contract in place work was completed and delivered can the person who paid the invoices try to dispute what was paid?

If they say the work wasn't up to spec, or if he says he was billed for something that wasn't provided, that's one thing. But if you did work as requested, and he paid for it, sounds like a contract was formed. You did the work, he accepted the work, he paid the invoices. The question is, did he receive what he paid for?

If he has evidence that he didn't, he can come after you to get you to refund for what wasn't done, or get you to do what wasn't done, or what was done but not in a workmanlike manner.

So find out what his beefs are, specifically. After you find out, you will know if you need to hire a lawyer, or whether you can just make it good.

I'm not licensed in Illinois so that's my two cents. What I said here is just information based on general principles of law, not legal advice. If you need legal advice, please consult a lawyer who holds Illinois licensure.