Hi there,
The requirements are quite clear and straightforward to implement. We can utilize a two level user system; admin and user(partner). Admin account can manage the users, add/remove accounts, allow api access, ect. We can offload api key generation to users as well. Each user would have an api key upon account activation. In addition, users would renew their api key themselves in case the key is compromised so that you(admin) wouldn't have to deal with such scenarios.
The API you've linke, clickwerx, is a poor design actually. It doesn't even utilize oauth which is like standard for decent APIs these days. It doesn't expose any endpoint to fetch, query the records either. Still, those may not be needed for your use case which is totally understandable.
Vertical parameter is not totally clear. I suppose it'll be treated as some sort of category/tag. Also, you've mentioned calls. I suppose these calls has nothing to do with this application which will be solely responsible for record keeping. Please let me know if I'm missing something here.
I am planning to use plain bootstrap 4 for the UI so no custom design. But it can easily utilize a new template if you decide to use a custom design in future.
I am planning to use php7.2 for the implementation. If you have a specific version in mind please let me know. It'll take a week to complete, thanks.