Microsoft SQL Server partitioning with on premise server
Yes this is possible, but is not a good idea. You can theoretically run a web application anywhere and punch firewall holes so that the web application can access a database on prem. This opens up a whole lot of issues as the database will have to be on a secure network segment and can wreck havoc on security policies.

You can make this more secure by using SSL for the database connection and use strict access lists on the firewalls, use Site2Site VPN and so forth but you are accessing the database over wide area network which can affect other users.

But in any case you will have to re-route all traffic from this customer to a different server, which will make the application more complex so it's probably easier to install the application on-prem for this customer.