In a 4-quart slow cooker, combine the onion, salsa, 1/3 cup of the vinegar, the brown sugar, cumin, chili powder, and 1 tsp salt. Add the pork shoulder and turn to coat. Cover the slow cooker and cook until the pork is fork-tender, 5-6 hours on high or 7-8 hours on low.
Transfer the pork to a cutting board. Using two forks, shred the pork. Discard the bone and fat. Put 1 cup of the juices and onions in a large bowl. Whisk in the tomato paste, the remaining 2 tbs vinegar, and 1 tsp salt. Add the pulled pork and stir to combine. If the pork seems dry, add more juices as needed. Mound the pork on the toasted hamburger buns.