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http://www.fvap.gov/pubs/vag/textvag/mn.html

E. Electronic Transmission of FPCAs and Ballots
Minnesota allows you to send the FPCA for absentee ballot request by fax. After faxing, submit the FPCA by mail
Minnesota does not allow you to receive the blank ballot by fax.
Minnesota does not allow you to return the voted ballot by fax.
Please refer to Appendix B for specific instructions when transmitting election materials electronically. Use the following numbers: DSN 223-5527, (703) 693-5527, 1-800-368-8683 (from U.S., Canada, Guam, Washington, Virgin Islands only). International toll-free fax numbers (See http://www.fvap.gov/services/tollfree.html ).

Minnesota allows you to send the FPCA for absentee ballot request by fax.
After faxing, submit the FPCA by mail Minnesota does not allow you to receive the blank ballot by fax.
Minnesota does not allow you to return the voted ballot by fax.

So there you go. Some states, MN being one, do not allow the ballots to be sent by FAX nor returned by FAX, only the application for the ballot can be by FAX (and in some cases by email or direct application through a state website). However if they, the state, and actually the local, usually county, authorities don't send out the ballots in time for them to make the round trip, or if they get "delayed" in the mail on the outbound leg, the military member is SOL. The law, IIRC says they, the states, have to accept ballots postmarked by the election, but IIRC, the military system does not provide date stamps on their postmarks, or at least they didn't in 2000, because that was an issue in the Florida vote then.