Multisite specific site registration


I am building a multisite where each "client" will have a site. The client will have users they need to add to their site. These users would have basic subscriber permissions. To make it easier for the client, I was thinking of creating a universal registration form on the main website that their users could use.

What I need help figuring out is if there is a way to create one universal registration form but maybe have a different site specific codes that each client to give their users. Or something like that.

And have them only registered on their appropriate site. I don't want them to be registered users on all the network sites. Just the one that the client gives them access to.

Any help or recommendations would be greatly appreciated!