Does Garbage Grabber work if I can't recycle cards?

No. Garbage Grabber's "Recycle 3 from your trash" is a cost, not an effect. Costs must be paid in full to activate the ability, so if you have fewer than 3 cards in your trash, you can't activate it at all (Rule 416.3).

This is different from recycling as an effect (like Dr. Mundo), where "do as much as you can" applies. But on Garbage Grabber everything before "Draw 1" is the cost — recycle 3, pay 1 energy, and exhaust — and all three must be possible to use it.