My contract ends soon, and the recruitment firm informed me that I was
offered a contract extension.
Am I capable of asking for a raise? Specifically, what is the norm for
raises in the case of extended contracts?
Yes, you can ask for a raise.
Contract extensions are fundamentally the same as a new contract containing all the previous terms, but with new dates. You are free to accept or reject such a contract.
In my experience, it's far more common to just continue the current rate. When I hire contractors, I often extend their dates and don't expect to pay more. (And what I pay isn't the same as the compensation the individual receives, of course.) But that doesn't mean you must follow the norm.
Presumably, you were offered an extension, but haven't yet accepted the offer.
In your case, you can say "Yes, I'd like to extend my contract, but I want to change the terms. I'd like to get $X." If the firm accepts, then a new contract is drawn up with the new terms and dates.
Note that your counter-offer could be refused, in which case you would have to decide to stay at your current rate, or leave.