What prevents a group of scientists from setting up doingscience.io or whatever, and publishing their results to a big shared blog? Do the contracts stipulate that their findings can only be published in specific paid journals?

Money. Research grants are given mostly by amount of publications in well known journals. If they just start a blog then they won't get further grants and have to change careers.

The effect of Impact Factor (https://en.wikipedia.org/wiki/Impact_factor) in academic grants is ridiculously high.

So in the end it's not the scientists who can make the switch. It's the people who pay the scientists.

Who are these people and how can people contact them?

These are mainly the administrators (read clerks) who are in the charge of the funding purse(s). You can contact them via your senators and other elected members.

Tell your representatives the following: The clerks that you have put in charge of funding our tax-money-filled purses for scientific research are just a bunch of uneducated or worse yet ill-educated fellows who do not understand even an iota of academics especially in the STEM areas. So they just rely on a bunch of very coarse proxy numbers (e.g. shitty impact factors) to decide funding. So please remove these clerks from these positions. We can and must install a better and more open system for reviewing where well known scientists/academicians can be invited to participate in the decision making process. Open access journals is one such great way to start. See what is being done by great academicians like Prof. Sir Timothy Gowers and Prof. Donald Knuth are doing for this cause [1], [2]. You may contact these people and get their advice in this matter.

Signed.

Your fellow citizen, a concerned taxpayer and a voter of USA.

[1] https://en.wikipedia.org/wiki/Timothy_Gowers#Elsevier_boycot...

[2] http://thecostofknowledge.com/#list

Your "clerks" are usually academics, actually.

I understand, but as academics become "administrators" and their goals shift, there is a non-zero chance of the academics metamorphosing (or gradually transforming) into "clerks".

Usually one must assign to the journal a more or less restrictive copyright to one's paper before being able to publish it. For math, this usually includes a stipulation that it be published nowhere else (except that many specifically allow posting to the arXiv, because—hurrah for the big guy being in the right!—it seems that no journal is big enough to fight the community consensus behind the arXiv).