In a large Dutch oven, combine chicken carcass and vegetables.
Add 8 to 12 cups of water, enough to mostly cover the bones and vegetables. Add vinegar.
Over medium-high heat, bring the stock to a simmer. Once the stock begins to simmer, turn heat down to keep it just barely simmering.
Cook for 3 to 4 hours, skimming fat off the surface occasionally, and checking to make sure the stock has not come to a boil.
Once the stock has simmered, and reduced by one-half to two-thirds, use a slotted spoon to remove and discard the vegetables and bones.
Strain the stock through a fine-mesh strainer and a couple layers of cheesecloth folded together.
Transfer the stock to mason jars, leaving plenty of headspace for expansion if freezing, and store in the refrigerator for up to 4 days or the freezer for 2 to 3 months.