I could be mistaken, but the way I understand it many people on MO are voting members of AMS, and combined together could ask it to change MathSciNet's current policy.

However, as others pointed out, MathSciNet works as a funding channel for AMS which would have to be either replaced by some other method of funding or leave AMS with less funds overall. A responsible proposal for making MathSciNet free would also somehow address the above alternative.