I've had an online account at AAA SoCal for years. There is a place to Make An insurance Payment (when you owe something on your insurance, that is), it's just that it only takes debit cards there. If you key in a credit card number, it figures out that it's a credit card (it must be coded in the number somehow?) and rejects it! If you key in a debit card number, it accepts it.

All the wording (except for one place) also says only debit, not credits, are accepted. (Some footnote or something one place says a "debit or credit number", which why I tried a credit card number, but credit card numbers don't actually work.)

Now, I pay my AAA insurance annually (since otherwise they tack on an 18% interest rate !), and I last paid in June, so I can't experiment with this myself again until next spring (when I again have some amount of insurance due).

Right now, it doesn't let me pay at all because it says I don't owe anything. But it does let me go to Manage My Payment Accounts (under my Auto Insurance), and there it says:
Every division of AAA is independent, and not only do they have different policies on insurance payment methods, they have different policies on lots of stuff. For example, at AAA SoCal I only get a passport picture at a discount. If I go get a passport picture at AAA Alabama while visiting relatives there, I can get it for free, because that's the policy of AAA Alabama, and for passport pictures the policy of where you're getting them applies, not the policy of where your AAA membership is based. But auto insurance is separate for each division, and thus you cannot pay the auto insurance you get from AAA SoCal on the AAA NorCal website, because it won't find your policy there.