Albert had it absolutely right, everything was about permissions, both access to the other computer and writing to the database on the local computer. Virtual directory was unnecessary. Whish I could do more than 1 up vote. Thank you again Albert!!

Earlier I had tried changing credentials but I was using the domain login rather than the local one. I needed [email protected].

I updated both the app's Advanced Settings, Physical Path Credentials and the application pool's Advanced Settings, Identity.

I also (as Albert suggested) created the same username on the other server and made sure it was in a user group that has access to the necessary folders.

With these changes my web app can now read from and write to the database on the other computer.