Wash the tomatoes. Dip them in boiling water for 30 to 40 seconds, or until the skins split and pucker, then dip in cold water or a bucket of ice and remove the skins.
Core and quarter the tomatoes, then boil them for 20 minutes in a large, uncovered saucepan. Run the tomatoes through a sieve or food mill.
Saute the onions, celery, peppers, mushrooms, and garlic in the oil until they are tender.
Combine the vegetables with the tomatoes, then add the rest of the ingredients. Bring everything to a boil, then simmer uncovered until the sauce is thick enough for serving, stirring often to prevent burning. The sauce should reduce by about half.
While the sauce cooks, prepare a pressure canner for hot pack, along with jars, lids, and rings.
Once your sauce is ready, fill hot jars, leaving 1 inch headspace.
Remove air bubbles and adjust the headspace if needed. Wipe the rims of the jars to remove food particles, then add the lids and bands and tighten until they are fingertip tight.
Load the jars into the canner.
Turn the heat to high, then allow the canner to vent steam for 10 minutes.
Add the weight, then process the jars at 11 lbs pressure in a dial gauge canner or 10 lbs pressure in a weighted gauge canner. Process pints for 20 minutes and quarts for 25 minutes. Adjust for altitude as needed.
When the processing time has finished, allow the canner to depressurize on its own.
Remove the jars, then allow them to cool for 12 hours. Check the seals before storing.