Yes, if it is an acoustic-electric fitted with a microphone or magnetic pickup.
Acoustic (acoustic-electric) guitars use different pickups and different amps than electric guitars. The characteristic of this setup is that the final sound output remains as close as possible to the original, that is, the "acoustic" sound. On the other hand, electric guitars are meant for tone shaping/coloring; hence, the amp is essentially an extension of the guitar itself insomuch as the tone is concerned.
This is to say: an acoustic guitar hooked to an electric guitar amp would sound bad, which is because tone coloring is an inherent part of electric guitar amps. It doesn't play well to varying degrees with an acoustic guitar. On the other hand, an electric guitar would sound very thin and almost tinny with acoustic amps because they are meant to amplify the original signal without altering it much.
With that out of the way and to answer the question, it is possible to use distortion on an acoustic guitar by using an effect pedal and an acoustic guitar amp. The amp itself wouldn't be able to add distortion on its own, though. The result would vary greatly depending on the pedal, the guitar, the pickup, and the amp. And you would probably need some trial and error to get a good distorted sound out of it.