No, it won't be a run-away query. Query execution is not logged to the tran log, many-to-many joins do not write to the transaction log of the DB (maybe of TempDB, but that's not the problem here)

Replication as a log wait means you either have replication configured or have CDC running and not completing.

Please read through this http://www.sqlservercentral.com/articles/Transaction+Log/72488/ especially the section on replication. If it's CDC, you need to identify what caused them to start failing and not allow the log to be cleared.

Changing the recovery model to full (or anything else) would not solve this problem, might make it worse though (you'd then need log backups to do what in simple recovery is done manually)

What you can do is temporarily add another log file which you can remove after the problem is fixed