Learn German: Wir haben Gluten nicht vermieden.

In sentences with a two-part verb (like the perfect tense here with haben and vermieden), nicht usually goes right before the second part of the verb at the very end. It negates the overall action of the verb itself, showing that the 'avoiding' is what didn't happen.

Since Gluten is a noun without an article, shouldn't we use kein instead of nicht?

You generally use kein to negate a noun without an article (like saying Wir essen kein Brot). However, using nicht here is perfectly natural because we are negating the specific action of avoiding it, rather than saying 'we avoided zero gluten.' When emphasizing that an action didn't happen to a known substance, nicht is placed after the object.

Could I change the order and say Wir haben nicht Gluten vermieden?

You could, but it completely changes the meaning! If you place nicht directly before Gluten, you are specifically negating 'gluten' rather than the whole sentence. It would imply 'We didn't avoid gluten (but we did avoid something else, like dairy).' To just say the action of avoiding didn't happen, nicht must go after the object.

What is the infinitive of vermieden?