The SQL server does not need to be shut down to do a backup. Actually, the SQL server can't do a backup if it is shut down.
I run backups everyday in the middle of the day. The server just takes a snapshot of the DB and backs that up, not disrupting access.
I would think your SQL server does nightly or at least weekly DB backups. These are needed to commit the transactions to the database.
Are you sure your system is not currently running backups?
you can check in Enterprise Manager \ SQL Server \ Management \ Database Management Plans or in Enterprise Manager \ SQL Server \ Management \ SQL Server Agent \ Jobs and see if jobs are currently set to run.
SQL 101 management rules are to run backups, optimitizations, and Integrity checks often; i.e. daily, weekly. I can�t believe your company does not follow these guidelines.
Then depending on how active the database is, you would also setup / run transaction log backups.