to cook food faster with

If the whole sentence is "A contrivance to cook food faster **with** is what I'm looking for" then I'd say the 'with' is optional there. The sentence has a clunky feel to me and might be better as "I'm looking for an appliance to cook food faster". For some reason, things like air fryers, food mixers and so on are generally called appliances.

Yes, it's optional. While the sentence is grammatically correct, it feels slightly awkward.

Just say "I'm looking for something to cook food faster."

I'll echo what others have noted here about the clumsiness of the sentence. To answer your question: no, "with" is not necessary and would be better removed. Unless you were going to say with which but again, no. The sentence is giving PowerPuff villain Dr. Mojo, honestly.