SQL application connection using one Windows user
If you're using IIS manager for application pools to configure identity for your application and your servers are Windows 2012 or later, you can look into using either a Managed Service Account or a Group Managed Service Account to run your app.  Either approach will eliminate the need to deal with passwords yet maintain a secure application login.

The comment provided by sepupic above is spot on though in that you should manage user permissions directly against the database separately, and how that's done will heavily depend upon what level of permissions are needed.

Hopefully this provides an approach to your situation though.