The relationship is simple.
The Vorta carry out the will of the Founders.
The Jem'Hadar do what the Vorta command.
If your race doesn't do what the Vorta command, they send in the Jem'Hadar. Then you die.
On a day to day basis, it seems so long as you meet your contribution quotas and don't cause any trouble, you're allowed general autonomy. The Karemma for example had a fleet and trade relations.