She can ask the court. However, absent special circumstances, child support ends when child turns 18 unless they are still in high school in which case it ends at 19.
Although there is no legal requirement to pay, there is certainly a moral imperative. The daughter should go to college and payment options should be discussed by you, your husband and the mother. Costs can be paid by combinations of scholarships, work study, student loans, parent loans, savings, etc. I am sure your husband loves his daughter and wants her to succeed in life. For that she needs to go to college.
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750 ILCS 5/513 governs college contribution. Yes she can petition the court, and it will go to a hearing if the parties are unable to come to a preliminary agreement. Because tens of thousands of dollars are at stake, it might be a good idea to talk to an attorney regarding his rights in this particular situation.