While Minnesota state law has a procedure that allows you to petition the court
for restoration of firearm rights under state law (https://www.revisor.mn.gov/statutes/?id=609.165), the problem you have is that a domestic violence
conviction prohibits you under federal law from owning a firearm. The only way to restore your firearm rights under federal law is to have the charge expunged from your record. If you no longer have a conviction on record, then the federal prohibition does not apply. Here is the information on obtaining an expungement
You may want to contact the court where your conviction occurred to see if you can obtain an expungement application so you don't have to draft the petition from scratch.
As you can see from the statute, expungement alone will not restore your firearm rights. You have to obtain the expungement for federal law purposes, and you have to petition for your firearm rights to be restored for state law purposes. Neither one on its own will allow you to own a firearm.
If the expungement or restoration petition are unsuccessful, you will have to obtain a pardon. The application is not online, so you will need to contact the Board of Pardons in order to obtain one. If and when the pardon is granted, they should let you know if you received the type of pardon that automatically restores firearm rights or if you have to petition the court for restoration following the pardon.
Minnesota Board of Pardons
1450 Energy Park Drive, Suite 200
St. Paul, MinnesotaNNN-NN-NNNNbr/>651-642-0284