Avis isn't at the Airport Blvd. location anymore; they are at the new CONRAC that's close to the 405 and no longer a walkable distance:
Avis moving LAX locations
Terminal 1 (Southwest); the first stop.
Use GasBuddy.com, the ZIP code for LAX is 90045. I use the gas station at Manchester and La Tijera (closer to the old Avis location) or the Vons on Manchester in Inglewood.