As of 2010, there were about 28,000 documented dues-paying Mayflower descendants. There have probably been about 16 million DNA tests taken by people in the United States, or about 1 out of every 20 people. Mayflower Society members may be more interested in their Ancestry than the typical person, so lets say 1 out of every 14 of them have taken a DNA test. Then there would be about 2,000 Society members who have taken DNA tests.
There were 102 men aboard the Mayflower who would be the ancestors of these 2,000 members with DNA tests, which is an average of only 20 DNA tests per ancestor. But let's say there was one person on the Mayflower who was prolific and has the largest proportion of the descendants, say 25%. Then that ancestor would have about 500 DNA tests we could use.
Now lets see if 500 DNA tests would be enough.
We each have 22 autosomal chromosomes from our father and 22 from our mother. At that level, if you consider each chromosome as segment, then we have 44 segments at generation 1. We'll take just the father or the mother, whoever is the descendant of the Mayflower ancestor.
Each generation recombines and divides the segments. The father's recombines less and the mother's more, but overall they average out to 34 subdivisions each generation. So at generation 2, you would have on average 22 + 34 = 56 segments. At generation 3, you would have 56 + 34 = 90. At generation 13, you would have 90 + (34 x 10) = 430 segments.
The 56 segments at generation 2 can come from 2 grandparents. The 90 at generation 3 can come from 4 great-grandparents. the 430 segments at generation 13 can come from 4 x (2 ^ 10) = 4,096 great(11)-grandparents. There likely is some pedigree collapse so let's say those segments come from 2,000 unique great(11)-grandparents.
The problem we have already is that a DNA tester has only a 430 / 2,000 = about 20% chance of getting even 1 segment of their one Mayflower great(11)-grandparent. So now we only have 100 DNA testers who actually have some DNA from that ancestor. We don't know which of the 500 those 100 are.
Also, the distance in centimorgans (cM) of the 44 chromosomes is about 6,800, so each DNA tester who got DNA from that ancestor would have only got on average
6,800 / 430 = about 15 cM in most likely just 1 segment, but possibly also in 2 or 3 segments. Those segments will be randomly distributed, but in total, we will only be covering 15 x 100 = 1,500 cM for all the 100 DNA tester descendants, which would at most only cover not much more than 20% of the Mayflower ancestor's 6,800 cM.
Some of those segments would overlap between DNA testers, which would theoretically allow for identification. But you must identify the DNA up through your ancestors one generation at a time with the help of genealogical records, ensuring that you have relatives supporting that the DNA came from one specific line rather than another. Techniques like Chromosome mapping and DNA Painting have only started to be used in the past few years to do this. And there are few people who have been able to map a line or two back to about 8 generations but not much more. That far back, both the DNA and the genealogical records start dying out.
Even with one line down from the Mayflower documented the 13 generations, you won't have the other lines of those descendants documented so that you could disprove that the DNA is from any other line.
So no, I don't think this project would be possible to do.
As far as Y-DNA and mtDNA for Mayflower descendants go, the only company that tests and gives Y and mt matches is Family Tree DNA. Family Tree DNA promotes group projects and one of the projects is a Mayflower project that is run by the Mayflower Society which currently has 1270 members. If you click on their "Project Statistics" page, you'll find statistics about the Haplogroups of the members, and you can find DNA details on the DNA Results and other pages.
Now I'm not saying that a project like this is not feasible at all. I just don't think there are enough descendants in the Mayflower pool and too few of the descendants will have enough of their ancestral trees documented.
However, this would likely be a very feasible project for Icelandic people. The Icelandic family tree includes 95 percent of all Icelanders born since 1703, along with half of those who lived on the island between 874 and 1703. You would need enough of the 300,000 descendants living today to take a DNA test so that most of them would have a few first, second and third cousins who also took a DNA test. So maybe 1 in 3 people would be needed, and then you'd be able to start assembling raw data sets for their parents, then grandparents, and continuing to work your way back. It would be very exciting if a project like you suggest got started for the people of Iceland.