export custom data (or xml) from app to phone?

If there's not too much data then you can add some fields to the fit contributor.

That would be an option. Typically, it would be a list of events: game minute - event - person.

Example.
2' : yellow home #13
24' : (1) - 0 #5
43' : 1 - (1) #12
67' : red away #4

etc.

Would that type of info be suitable for a fit contributor?

It's not easy. On one hand you want to have as small (in bytes) fields as possible, because I think you can have 32 bytes together, on the other hand it's not some kind of record. You can try to do it in some separate fields, but it'll not be very user friendly (in the Garmin Connect) to look at it.

You can try to do it as a string, similar to how you wrote above. I guess the time is not needed, because it will be automatically added.

The other question is how you do this. Record or maybe Lap type?
Maybe as record, write it, wait at least 2 seconds before you write the next one (if 2 players get yellow card at the same second, then you can't report both in the same second, as the 2nd will be only written to the fit file, you'll need to make it sure it's written out before you sent the 2nd one, that's why I wrote to wait for 2 seconds) This way the data will be part of the fit file, though still probably not visible in Garmin Connect (but available for other platforms)

The only way I can think of making it visible in GC is to do laps. For the purpose of me explaining it you could do laps for every time interval when they're playing. When something happens then you finish the previous (current) lap, creating a new lap, in which you report the event, then start a new lap. So you 'll have something like:

lap 1: 0:00 - 2:13 - game
lap 2: 2:14 - 2:54 - event: "yellow home #13"
lap 3: 2:55 - 24:16 - game
lap 4: 24:17 - 24:18 - event "(1) - 0 #5"
...

This way you'll be able to see the string field in the GC's lap view

Thank you for the thourough explanation! Yes, it sounds like its not that easy and not that straightforward. But maybe laps is the best way. I would need to rethink my app a bit, since each period is its own lap. But that might be possible to overcome.