Can you attack with a Vehicle the turn it enters the battlefield if you crew it immediately?

Vehicles enter the battlefield as artifacts, not creatures. Summoning sickness (CR 302.6) only matters when a permanent is a creature and is being asked to attack or use a tap ability. Because a Vehicle enters as a non-creature artifact, it does not 'have' summoning sickness in the traditional sense while it is an artifact.

When you crew a Vehicle (CR 702.122), the Vehicle becomes an artifact creature until end of turn. The key rule is CR 302.6: a creature cannot attack or use tap abilities unless it has been under your control continuously since the beginning of your most recent turn — but this check applies at the time it tries to attack. If the Vehicle became a creature (via crewing) during the same turn it entered, it has only been a creature very briefly, yet the summoning sickness rule tracks control of the permanent, not how long it has been a creature.

CR 302.6 states the permanent must have been under your control since the beginning of your turn — it refers to the permanent itself, not to when it became a creature. Since the Vehicle entered the battlefield this turn (i.e., was not under your control since the beginning of the turn), it is affected by summoning sickness and cannot attack the turn it enters, even if crewed.

The creatures used to crew the Vehicle are unaffected — they just need to tap, and tapping to crew is a cost, not an attack. The crew ability itself works fine. It's the Vehicle that cannot attack due to summoning sickness.

Example: You play Smuggler's Copter on your main phase, then tap two 1/1 tokens to crew it. Smuggler's Copter becomes a 3/3 creature — but because it entered the battlefield this turn, it cannot attack this turn. It will be able to attack on your next turn if it survives.