You can go onto their website and for a small fee submit your proposed line which someone will research and see if they can find a matching documented line. This will give you the information for how far down the line has been proven and give you a starting point for where you would have to provide additional documentation.
The following is information sent to me in email after a successful proposed lineage search.
"In general, an applicant will be required to prove their lineage beyond whatever point for which the Mayflower Society already has an up-to-date previously approved lineage paper on file. Usually, the best place to begin working back in order to link into an accepted lineage is with your immediate family. Full documentation, including all applicable birth, marriage and death certificates, for the most recent three generations should be assembled. When moving back beyond grandparents, documentation can be provided in the form of birth, marriage and death certificates, wills and other probate records, land records which state relationships, or, in the absence of these, published sources such as vital records of towns, town histories, published genealogies etc. Census records may also provide good corroborative evidence. There are a number of other types of records that can be used to prove descent from one generation to the next, but this is the challenge of proving a Mayflower lineage: finding these. Hopefully you already know some of this and perhaps your family has some of these documents."