This New York Times story explains the difficulty of collecting signatures for Republican primary candidates in New York city. With only 470,000 registered Republicans in the city, even mainstream Republicans have difficulty collecting 3,750 valid signatures of party members. As the story says, normal petitioning on the street barely works, and petitioners find they must first get a list of registered Republicans and then try to find them at home.
This is why the new legislation in Arizona, HB 3605, is so devastating to Arizona’s minor parties. The bill requires over 5,500 signatures of party members (or registered independents) and the Green Party only has about 6,000 registered members in the state.