Need to update user's Org....somehow

I would appreciate any guidance I can get on this. We serve students and teachers in high schools and colleges across the country. We need to associate all of our users with an Organization (for a long list of reasons). We use domain matching whenever possible, but it doesn't work with gmail, yahoo, and other popular email domains, because they could be from any Org.

We currently have our users fill out Zendesk forms where they tell us what school they are with. But I have never found a way to get that information to the user's account for future use, other than doing it manually. This is a huge time-waster and it's getting worse and worse on our end for a variety of reasons. I really need to find a solution.

I have used webhooks/targets with triggers/automations (as an example, I can update a user's tags based on their choice of personal pronouns) but I haven't figured out how to apply that to the Organization.

Generally speaking, I can see something like having a dropdown list of Orgs for them to select from, which in turn adds a tag to their user account...but how do I get that tag to actually associate their user account with an Org? This seems like a huge missing piece of the Zendesk user profile puzzle, but I've never found a way to get this to happen without manual intervention. If I'm missing something obvious, I'll be happy to hear about it!

I am not a master programmer, but I know enough to get around and I'm more than willing to learn. I am open to creative ideas even if they take some effort to implement up front, as the long term gains will be worth it. 

Thanks in advance! (And apologies if this is in the wrong topic....there are too many to pick from!)