Another success. I'm going to assume that this is completely fixed, so I'll "Solution" this thread.
If it's a regular bank card, as in connected to a checking or savings account, then I believe so. Don't take my word for it, though. The option to pay with debit is there, so it's probably available for automatic, too. With that, it's really no different than paying with the checking account itself, which I know they accept.
Regarding prepaid cards, though, I didn't ask the rep if I could set it up for automatic payment using it that way, though I think if that would have been an option she would have suggested it. She seemed to have more knowledge about the situation than the person I spoke with last month, who suggested it was some screw up regarding the credit card system. It was at least a relief to finally find out what the actual problem was.
Edit (7/1/2018): Same reason as listed in original post.
