You might start small and try to get the Gates-1183 and ancestors' profile opened up. I messaged the profile manager a few days ago with no response as yet. (Deceased notable ancestors should always be open, and he has no living ancestors; also many died over 100 years ago.)
As for DNA, his lineage has been traced, and it has generally been imputed to be at least R-M269 (not terribly helpful), yet eupedia claims a "close relative" tested I2-Y3684. I'd expect it would be difficult to get any very close relative to publicly test.
So besides opening up his line on WikiTree, the best you are liable to do is compare others who share a common line on paper. My own 2nd ggrandmother on my maternal line is Gates-3176, and I believe her paternal line is the same as Gates-1183 (whether it is biologically is basically unknowable, without an actual test from him, though testing descendants on different lines can at least prove the type of the common ancestor). From autosomal matches I have identified other direct male Gates cousins and I could locate others who have not yet tested. I would be interested in getting at least one of them to do NGS.