Yes...the donor must be a male with the surname of Davidson/Davison/Davisson (or the "blood equivalent," if the donor's surname is different for some reason.....like an adoption, for example), in order to join that surname test. There are MANY surname websites/testing projects, however, and there are also a few different companies that perform the actual testing.
Note: Some surname testing projects rely just on the testing company's own website to post the results and all other information. The Davidson/Davison/Davisson project (like a lot of projects) however, also uses the separate "stand-alone" website www.davidsongenes.org
so that we can present things in our desired format, arrange matching/nearly matching donors into "family groups" and so that the donors have more flexibility to enter information.
You could perform a general internet search using something like the following: "Davidson DNA testing project" (but substitute your surname for "Davidson") and see if you get anything. There may be a specific stand-alone website that addresses your surname. By the way, I believe that Ancestry.com bought the testing company named Relative Genetics, and the Davidson/Davison/Davisson project uses Family Tree DNA (FTDNA)......as do a lot of surname projects. FTDNA has hundreds of surname projects, and if you go to their website, you can see if your surname is included (and if it is, that website will probably give the e-mail address (and/or website address) of the "project coordinator" who oversees that surname).
FTDNA (and other testing companies as well, I assume) will "estimate" your haplotype from your marker results, but there is also a separate/extra test where they will actually "directly determine" your haplotype (this may be one of the other two tests that Ancestry.com mentioned). The haplotype can help to determine your family's heritage, going "way back" to when folks were first starting to "roam around the earth" (i.e., I have a haplotype of "I," and that supposedly means that my Davidson line can be traced back to the Vikings). The actual markers, however, are what you really need to see if you match/nearly match another person with your surname. FTDNA offers 12, 25, 37 and 67 marker tests, but whatever company you choose, I would not do less than 30 markers. Some surname websites will not even post the data unless the donor did 30 or more markers (doing fewer can easily lead to "false positive matches"). It is less expensive to join the Davidson/Davison/Davisson DNA project if the donor goes through our stand-alone website (this gives a "group rate," as I understand it) than it is if a donor goes directly to FTDNA on his own. I don't know if something similar is true for Ancestry.com/Relative Genetics.
Also, if you know any male Davidsons from your mother's family, you could see if one of them is willing to be tested (and you could offer to pay for it). That type of thing happens all of the time. Good luck!