Does a full transaction log cause a rollback?
Yes but it will be the rollback which actually causes the log to become physically full.

As a transaction generates log records, it will also reserve space in the log for any records required required to rollback. With 20MB of log space remaining and a hypothetical transaction that generates 10MB of log records, you would receive the log full error with 10MB still free and the rollback would then consume that 10MB.