Technically, this is possible but it is not a solution VON can recommend or support. Microsoft has information on connecting to Express from another computer here:
As you can see from the Microsoft article, there are additional security concerns with this model and additional configuration required to implement safely. Vermont Oxford Network has limited resources and cannot assist your IT staff in implementing this sort of solution.
Vermont Oxford Network recommends using an existing Standard or Professional instance of SQL Server 2012 or better. If you do not have a full instance of SQL Server available with DBA or similar IT professional support, the next best alternative is to install SQL Express locally for use on a single PC.