This issue sounds like it may require more troubleshooting than could be provided here on the forums.
I would strongly encourage you to send a Support ticket so you can talk to a Tech Specialist directly. They'll be able to work with you to resolve this issue.
You can contact them here: https://support.leagueoflegends.com/home
In the meantime, if you keep crashing due to a bugsplat message don't try playing a Queued, Ranked, or Co-Op vs. AI game. If you keep crashing from these you will gain a Leaver Penalty, and eventually the Automated Leaver System may place a temporary suspension on your account.
If you need to test a possible fix, you can always launch a Custom match. The Leaver System does not apply to Custom Matches.