Voting methods are mostly set state by state, as I understand it. (I'm pretty ignorant about politics.) Some states, like California, allow powerful ballot propositions. Thus you could get a change like this on the ballot, starting in a few states, without going through the major parties. In California it doesn't even seem that wild to hope the parties would support the change: in recent years it changed to "open primaries", and I think it'd be rational for the D's and R's to want approval voting in the primary. You get experience with it and it starts to become thinkable elsewhere.

Or start using it in nongovernmental orgs, and likewise gain experience and familiarity.