Without a context, the meaning is:

I did something that made the plane fly.

The chance for a plane to fly compared to the chance for a plane to be stolen essentially rules out the second interpretation.

Even

J’ai fait voler la voiture.

would more likely mean the car took off for a moment because of you.

To make the stealing more plausible, you might say the ambiguous:

J’ai organisé le vol de l’avion.

and to make clear it is about stealing:

J’ai fait ce qu’il fallait pour que l’avion soit volé.

It is ambiguous if taken out of context; it can have two translations.

• 1/ I made the airplane fly.
• 2/ I had the airplane stolen.

In « 1/ » the action can be one several sorts. It can be the supply of the necessary maintenance and fuel for the plane to be able to fly; It can be a repair that was needed; it can also be the exercise of particular piloting skills (for instance in a damaged plane that was hit while flying); in the case of scale models it can be the action of remote control piloting; today it might be used for remote control piloting of piloteless planes.

It can’t be "I have flown/flew the airplane." nor "I had the airplane flown.".