Can I give other players permission to use my Buggy or Copter?

No — there's no way to share driving. Only the player who crafted the Buggy or Copter can drive or pilot it.

What others can do:

• Copter: its passenger seat is open to anyone, so a friend can fly with you.

• Storage: on PvP servers anyone can open your Buggy's boot or your Copter's storage. On PvE servers only you can.

Cars found in the world are different: they have no owner, and anyone can drive them.