You need to sign in with a Microsoft Account to use the Mail app.

If you have two accounts (one Microsoft account and one Non-Microsoft account) associated with your Mail app, and if you try to remove the Microsoft account (to be left with only the Non-Microsoft account), the option that you get is Remove all accounts.

So to answer your question, you can NOT use the Mail app without first signing in to a Microsoft account. You'll get the Accounts option once you add a Microsoft account.
After you add a Microsoft account, the Settings charm will have an Accounts option, which you can use to add other Mail accounts.
You can however, turn off notifications for the Microsoft account, and keep notifications on for the other account(s).