Hi there! I've already integrated zoom video conference for a telemedicine application including a scheduler module and I can tell you some gotchas -
First, the zoom meeting will open in an external tab initially and continue in the zoom software. So, as a prerequisite all users must have this installed.
Second, you absolutely need to have a business plan with them as the API is not available for free accounts. Ideally, the number of hosts equals the number of teacher in your case. However we were able to optimize to half the required hosts by using a few optimizations and techniques. May or may not workout for your case though.
Finally, there are other options like tokbox which give you an "within" the app video calling but with ASP.net, my assumption is that it's not a SPA, so might be an hassle as every page change means disconnection and connecting back. So, unless you are using angular or react for frontend - or planning to migrate - zoom is indeed the best bet.
Ping me if interested to push and close the project asap. Best of luck in any case. Laters.