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Take some scissors, walk to your nearest park, and start cutting the nets?
Seriously, though, I think Open Broadcasting Studio has ways to record your screen, and you should be able to just crop from there? (Someone with more knowledge here please chime in)
Probably the easiest way to do it real time (still not particularly easy) would be to have OBS capturing the feed, have the instant replay buffer enabled and when a goal is scored, save the buffer, maybe trim it with a simple editor (or even better, just ffmpeg) and then upload the clip.
I'm sure there's better ways, but when I've done it I've just made a screen recording, like with QuickTime or whatever.