None. We only had our permits for the cables. Before we started our project I contact Yosemite in regards to what permits would be needed, but it was too late to apply for one and have it in time for our trip.
You're also not allowed to bring motorized gear into wilderness territory--which our dolly is, but very small and quiet. The ranger never asked/told us we couldn't have it. I guess its probably luck of the draw. I wouldn't suggest doing what we did, but it worked out--I can't say everyone will though.