Nate,
DI schools, by rule, are prohibited from having any sort of tryout. I'm not sure why this would be the case, but that's the way it is.
DII schools, along with JUCO and NAIA schools are able to individually work out a player to determine their suitability for the program. I don't know what the rule is for DIII.
Because of the scholarship limitations, there are many walk-ons at DI schools, but chances are they are not quite the same as you are thinking. They are recruited walk-ons that the coaches have seen play (or had recommended to them by sources they trust) and they are just as much a part of the team as the scholarship players, they just do not receive any athletic financial aid.
The "true" walk-on is probably rare, but it does happen. Just know that the cards are stacked against you and you would really have to stand out in order to make the cut.
Does anyone know the rationale for why DIs are prohibited from working out an individual player?