Are you talking about the skilled evolution? Synth in general wouldn't get any bonus to crafting from just having the eidolon.
If he does have the skilled evolution, I don't see why not. You can rule it as being impossible due to the lack of hands, but seeing a quadruped nom on some raw materials until they craft something pretty effectively would kind of make me laugh personally. So, YMMV.
If he takes the evolution to allow his form to have a bonus to crafting, then he has that bonus because he has evolved a way to do it, perhaps his claws become retractable. HOW he gets the bonus is completely flavor text.
This hardly seems like bending the rules, and more just using the abilities the class has.