Is it possible to automatically generate highlights from video?
Theoretically, sure, it is probably possible to devise a computer program that could recognize the ball entering the goal and cut video around it, however this would have to be custom computer vision code and not something that would be generally available to the public.  

It probably isn't as simple as just a motion trigger either since it needs to be something just the right size moving.  It would also be exceptionally hard to tell a goal attempt apart from a goal, though perhaps you could have it watch the net for movement using security camera software or something related assuming that the net is otherwise pretty static (and depending on your angle).

Overall, I wouldn't expect the greatest of results from such a system though.  It sounds problematic at best, even if you can get it working in the first place.

If you can synchronize a clock with the start of the recording, your better bet may be to hit a button during the game to take note of the time each time a goal is made and then try feeding the list of times in to a scripting based video editor that can cut 15 seconds before the goal and a few seconds after from the video and put them all together.  It's probably less work to get configured and more reliable, but it still would be a fair bit of work to do.