I think it takes 25,000 signatures. The constitution party is running candidates and is currently collecting signatures. It has a lot of slots open.
http://www.constitutionpartyil.com/
In Illinois, it takes 10 signatures to get on the primary ballot to run as committeeman for your precinct. This according to Doug Ibendahl, co-founder of Republican Young Professionals. But there are different rules for Cook County.
Last fall there were thousands of open slots for the Republicans. The terms are for 2 years. Further details @ Republican Renaissance Project.
I personally do not think that a third party can get the job done quick enough. But I do encourage those who want to, to do what they feel is best.
You do not start building a house at the top. You need a foundation. The committeeman is part of that foundation.