How much power can a 7805 voltage regulator dissipate without a heatsink?
No. Even if the output transistor did not saturate and even if you had a PERFECT, heatsink of infinite size and infinitesmal thermal conductivity, you could only manage 2.85A due to the junction-to-case thermal resistance being 5K/W at room temperature. So in reality you wouldn't even get 2.85A is unobtainable.

Datasheet lists peak current as 2.2A which is just that: a peak current, not continuous. Since there is no input voltage given, that's also probably right at dropout (i.e. minimum heating) and 12V to 5V is well beyond that so you would get significantly less than even 2.2A