Webforms module breaks MySQL database
I would expect this is more related to the backup and migrate module.

The addition of the webform module's data may have pushed your setup over its limit when doing large amounts of processing like backup and migrate would be doing.

I would suggest checking your MySQL configuration's max_allowed_packet setting, it may be too low.

I would usually recommend setting it to 64M for Drupal 7 and then restarting the MySQL service and see if that resolves the issue.