Using Mixed mode authentication to authenticate Active Directory users to an application
Having an application connect would be very different than an end user on SSMS.  Applications tend to use connection strings.  Your applications connection string or config might need to be modified to connect to the SQL Server as an AD user as opposed to putting in a username/password for a SQL user.  

You will want that user to have the same rights as the SQL User and probably want it to own the database.  Contacting the vendor might be a good case.  Going with AD users gives you more auditing visibility anyways which should please auditors.

Some connection string examples:
https://www.connectionstrings.com/sql-server-2012/