Even if they don't have phpAdmin they must have another tool like it within the Main Website Control Panel. The same place you would have been when you created the original database.

Either there or another location within the Website Control Panel (not the UBB control panel) and find one that does database backup. You should also find one that backs up the entire Website which you should run once in a while as your host provider may not be backing up your data, often it's up to you.