Change database based on request origin

I am developing a web application which will be run by multiple companies. Each company will have its own instance of that web application. And each company has its own database (the web applications and databases will be running on the same server).

Each of these instances should be talking to one single Web API application to retrieve data from their database for that specific company.

Now I am wondering whether it is a good practice to change the database connection string in the Web API application based on the request origin? Or am I overlooking something?