You mail the returns to the IRS or the state no matter what mailing service you use. Not sure what you think it has to do with PO boxes. If you do not want to use the U.S. postal service for some reason, then try UPS or Fedex. There are a lot of ways to take your envelopes to them--try Staples, Kinkos, UPS, etc.
For federal returns: https://www.irs.gov/filing/where-to-file-paper-tax-returns-with-or-without-a-payment
For where to mail it to your state look on your state's web site:
<a rel="nofollow" target="_blank" href="https://www.irs.gov/filing/submission-processing-center-street-addresses-for-private-delivery-servic...>
I think this (which I found in the meanwhile) partially answer my own question. But I am having trouble finding the analogous information for Illinois.
I asked because I am mailing internationally.