Learn Swedish: Jag känner ingen som aldrig äter gluten. - I know nobody who never eats gluten.

This is because of the BIFF rule. The word som starts a subordinate clause. The BIFF rule tells us that in a subordinate clause, sentence adverbs like aldrig, kanske, or inte must come before the verb. In a normal main clause it would be äter aldrig, but here it must be aldrig äter.

Could I use vet instead of känner for the word know?

No, because Swedish has two different words for know. You use känner when you know a person or are familiar with a place. You use vet when you know a fact. Since you are talking about knowing a person, you must use känner.

Is it possible to say inte någon instead of ingen?

Yes! ingen means nobody, and inte någon means not anybody. Both mean the same thing. Jag känner inte någon is perfectly correct, but Jag känner ingen is very common because it is shorter.

Does the subordinate clause som aldrig äter gluten have a subject?