You typically don't want to restore a database on a live system when there are users accessing it. However, this particular scenario makes it sound like this database is not in a multiuser environment, and when you are ready to restore the snapshot, then no one is using it (specifically, no one is adding vital data to it).
If I understand the scenario correctly, you can use mysqldump to dump the database "snapshot":
mysqldump -u username -p password database_name > snapshot.sql
At the end of the day, it can be restored just as easily:
mysql -u username -p password database_name < snapshot.sql
Your web designers were likely thinking of restoring by saving the actual datafiles and then replacing those. In that case, they would definitely want mysql shut down.