This question confuses me. It seems akin to asking "can I have a dragon that's a monstrous humanoid?" A spider is an arachnid, which is a vermin. The rules don't specifically say "spider can't be animals" because, IMO, it'd be like saying "dragons can't be monstrous humanoids." They have a type, and are bound by the restrictions of that type.
That said, if you're the DM in this instance, you can break those rules at will. If you were a player in my game, I'd give you an odd look, then suggest either a magical beast (augmented vermin) or animal (resembling, but not, a spider).
If you don't want a magical critter, aberration works as well. Mutant spiders that for some reason are a bit smarter than vermin?