Much like how Spock doesn't punch people to knock them out, it seemed to me that George isn't really the type of character that would use a blaster. We learn later that he has his own ways to take people out, but for the time being, I think his inability to use a a blaster lent him an air of innocence.
Now, that's not to say the he doesn't have a blaster. As the Author indicates here, all of those Megaman-esque suits come with a blaster built-in. This comic was my explanation, one of many possibilities I read about online, of where Mega Man's blaster goes when he's not firing it and why it seems to randomly appear on either arm depending which direction you're pointing. As the Author explains, you just have to know how to make the blaster form out of the suit, which George apparently doesn't nor does he ever really need to learn. (Well, I guess there was that one time.)
By the way, did you catch my half-assed explanation for how Bob destroyed Mega Man's "force ghost"? Once again, not a clean explanation, but an explanation nonetheless.