I had posted about this on the WDWinfo site too.
While the trams (at least at WDW) operate all on private property, I am only guessing here, they probably do not need any license at all. Even when I have seen them moving trams from park to park, it is always going to be on "Disney Roads". They are never leaving the property.
My wife is a school bus driver. I know that to operate a bus, you need a CDL, and you need a "P" (passenger) endorsement. To operate a school bus, you also need an "S" (school) endorsement. To operate an articulated bus (basically anything with a joint that combines separate vehicles/trailers), you need a Class B CDL.
While it is true that these are all on private property, from a liability point of view, I am thinking that if there was an accident (imagine someone getting hit by the tram), that Disney would probably prefer to show their drivers had the "Proper Training"...
I could see the lawsuits now. Not that they wouldn't happen anyway, but imagine the headline "Unlicensed Disney Tram Operator to Blame in Crash that Injured..."