“Live local.” Is it a complete sentence?

It's worth mentioning that in "live local" (like in "think different"), "local" is an adjective that's being used as an adverb, ie "live locally", "think differently."

I guess it's like "Think different." I don't have an answer but I would be curious to know if "Live local" would be grammatically correct.

Yes, it's a complete sentence. It's an imperative, urging you to, perhaps, participate in matters relevant to your local community and/or stimulate your local economy. I don't know your context, but I imagine it's something like "Don't travel to the Riviera; go to your local beach. Don't buy cheap foreign knock-offs. Forget the Louvre; go to a nearby museum. Eat at your local restaurants. Join the PTA. Live local."