In NM you need ID to register. At the polls they ask some PII. So there is room for some fraud. Significant numbers would take time but I guess it's possible.
First gun I bought was the Winchester 30.30. Going out of business sale in Columbus GA. $59.95 cash out the door with two boxes of shells. Still have it. This was in 1977 BTW. I like carrying it in the woods better than my Remington 700s.