Okay. Firstly, that's the first Leatherworker quest, not Tailor.
Secondly, yes, Leatherworkers and Tailors get the Leather and Cloth they need from Disassemblers. (But not the twine or thread, that's from the Trainers. Lace and buttons and buttonhooks also come from Disassemblers... but that's later.)
However! When you accepted this quest you were given 20 Damaged leather thingies to make bridles with. Simply buy the Leatherworking Tool from the Leatherworking Trainer, and use it to start Leatherworking. Click on the Reagents Tab of your (B)ag and move the leather into the window and start and get more positives than negatives and so on and so forth until you run out of leather and understand how to stitch things together. Then turn in the bridles.