My employer didn't withhold state taxes and now I owe. Is there anything I can do?

No. Either way, you would have paid the same amount. This way, you just pay it later. If you owe an underpayment penalty; it's not much more than the interest you coulda earned on the money you had use of.

Better give your employer a new W4 or ask them to start withholding for the state.

@Hal_Al - have you seen the penalties Illinois charges.  2% if no more than 30 days late 10% if more. .

I use to live in IL, but it's been a while. My comment is  more about "under payment" penalties as opposed to "late" payment penalties.