Should I just move on if a professor doesn't answer my emails requesting a letter of recommendation?
Stop by his office (if in town) or pick up the blower.  Just ask him directly if he'll do it or not.  A lot of people are not so email/text centric.  He'll either say yes or no, but at least you'll know the answer.  (And if he says no, just drop it and move on--don't argue.)  

Personally, I wouldn't cross off and move on, just because of not responding to email.  Especially if he's the best one.  He is probably just busy.  Or a procrastinator.

Voice or face to face are direct.  An email is like a letter.  Easy to ignore, to get to it later (which becomes never), etc.

When you do confront him if he says, he's unsure of what you did, offer to give him some bullet points and a BRIEF explanation of what you're going after and key features to emphasize.  (In the nature of making his job easier.)