MN is a Standalone state (like CA), which means transmission of state returns does not go through the IRS' system, which has been shut down.  Technically, you should be able to e-file.  When you try (and make sure you check the box to only e-file MN), did you get any diagnostic or error?

https://proconnect.intuit.com/community/individual/help/ef-atlas-federal-state-ef-method-standalone-...