Although your ssh login works without a password, you also need to configure sudo on your Target server so that it doesn't require a password.
Since you only need for the master user to be able to run commands as the appmanager user, you can handle this completely through sudo. In /etc/sudoers
on Target, add:
master ALL=(appmanager) NOPASSWD: ALL
Then your ssh command becomes:
ssh -t master@server 'sudo -u appmanager /app/acme/stopApp.sh'
If you need environment setup that is normally handled by the appmanager login shell, you'll need to source that setup explicitly. (And I would argue that the deployment environment setup should not be confined to a login shell anyway.)
More details on how to let one user run commands as another user are in a ServerFault answer and the much denser sudoers man page.