I reported a road kill deer, then printed the form in which I sign. Is that my permit? Is this all I need to do?

Yes, if you are talking about "salvaging" a road killed deer.  Within 24 hours, you can report a wildlife salvage by going online to fishandgame.idaho.gov/roadkill, which will direct you to a self-generating permit that you can print. A copy of the permit must accompany the salvaged wildlife.