Legal Question in Real Estate Law in Illinois

Yes. Taxes always come first.

There is a clause in either the note or the morgage or both that allows for the bank to make this payment.

Look for a paragraph called something like "Application of payments" on your mortgage. It will specify how late or short payments will be applied. Most allow payment of taxes and insurance before principal.

As per the other answers standard lender documents do have such provisions. However, what I've seen is that some lenders are willing to let property taxes go "somewhat" delinquent since they would rather have the P&I performing. However if you have a tax escrow they are responsible to make the tax payments or you could complain even if you are delinquent on the P&I payments. If you waived escrow or a pledged account, then their paying taxes may be a bit unusual, but allowed assuming the loan documents allow it.