Yes it does. & the FP machines are located before you get to the ride itself, saving you a few steps & avoiding some of the crowd at the entrance. (It's on the right as you walk towards Big Thunder Mountain RR).
The BTMRR FP machines are often down during slower seasons. I would bet they are used for Summer, Thanksgiving, Christmas, Veteran's Day, Spring Break, weekends, and major events. I don't think they even use them during Presidents Week.