You can add an obligation to multiple Stewardship Sites at once with a Stewardship Site view.
Start with a Stewardship Site view:
Select the records you want to add the obligation to, then click the 'edit' button. You can select all the records in a view by selecting the topmost check box.
Select 'Add obligations' from the available options, then fill out the obligation details. Click 'OK' when you're done.
Check out this article for more information on Stewardship Site Obligations.