Is it safe for a provider to accept credit cards? Is that even done? Who does it?

I don't want any record of the transaction,  like it never happened. Could be hard to explain. Possible IRS issues too

It can certainly be safe. I've done this with providers several times. Many have registered side businesses and put up websites for this purpose (e.g. massage services, wellness services, sex coaching therapy, antique dealer, etc.) However, you'll probably find this more common within the Vegas provider scene.

You can search Tryst and filter by keywords like CC or credit card and then reach out directly, just make sure you cross check for reviews on TER because of obvious scams.