Am I legally allowed to accept fiat currency online and return a counterparty asset/token?
It depends on where you live, where you sell and who the buyer is.

If your jurisdiction regulates this stuff or your buyer is say, a US citizen, regulations may apply. You may be able to work around that by implementing certain basic rules such KYC/AML, either full blown, or basic (such as the user confirming they're not a US resident or citizen, etc.)

There are several games that do exactly that on the Counterparty platform without issues, but that doesn't mean anyone can do it.

Recently there's been a growing number of regulations about ICOs and "tokens", so best check with someone who understands laws and regulations that apply to your territory and is qualified to give legal advice.