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 unless you lock it, 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.

Locking your Copter. Walk up to your Copter and tap Lock. While it's locked, nobody else — teammates and clanmates included — can take a passenger seat, open its storage or refuel it; they'll see "The owner has locked this vehicle". Someone already sitting in it can stay until they get out. Tap Unlock to open it again.

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