You could use Acra. I have used and it is great. It can send the crashreport to a GDoc.
I’m pretty sure there are other tools too, and I think since Android 2.2 google has something built-in but I haven’t found any docs about it. (Android Error reports).
I would recommend Bugfender. It lets you send all logs to a centralized cloud server, completely managed by you, no need for the user to intervene.
Note: I am one of the creators of Bugfender, still I think this could help solve question.
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