You can accomplish this by modifying the HTML code (link) you are using on your webpage to open the form, for example with a "Click Here to Register" link.
The basic HTML code starts here:
Where xxxxx is your Organization ID.
To force the form to open for a specific Location you will add &Loc=locationcode to that HTML.
If your Org ID is 522358 and you have a Location named Jackrabbit Education with a Location Code of EDU, you would add &Loc=EDU immediately after the last digit of the Org ID.
The link code would be:
This will auto-assign the registering family to the Jackrabbit Education Location as well as only show classes in that Location to choose from.