I live in CA. I apply at the national website, in Plymouth. You fill out the little "Preliminary Review Application" first, pay $10...and wait about a week. Someone emails you telling you what they can and can't confirm in your lineage...and then they tell you that you can purchase, for another $10, their approval paperwork based on an established "line". Each line has a codenumber, and only exists because someone in the past has requested the same proof as you.
So, if you have the top 5 or 6 generations right, they will tell you, you are approved for Line #____, up to the 5th generation. Then you take that paperwork they give you, and take it to your STATE Mayflower board. Each state has their own website.
When you apply at the state level, you cite the Line # that you are approved under, and then provide supporting documentation from the point that Plymouth left off (like the 5th or 6th generation)...with a new application, new fee.
If they approve all that, they will give you a new Line #, and a certificate declaring you a descendant of __ from the Mayflower.
I've also applied and gotten told that my proposed lineage is wrong at the get go...like they can prove me wrong with docs they have on file. This is at least useful to know, so you can correct your tree.
Hope this helps.