If you're flying into Cozumel airport, the closest Budget car hire location to you is Boulevard Aeropuerto Cozumel, and they can be reached on +52 987 688 1241.
No, Budget does not currently offer pickup services from Cozumel airport (CZM). However, their nearest location is Boulevard Aeropuerto Cozumel which is approx. 0.337mi away from Cozumel airport and accessible by public transport.