Wash all vegetables under cold water. Drain.
Peel the tomatoes by blanching them for 30-60 seconds in boiling water, then transferring them immediately to hot water. Remove the peel by sliding or cutting it off, then core the tomatoes. Measure out two quarts and set them aside.
Peel and cube the potatoes (½" cubes). Measure out 1 ½ quarts, then set aside.
Peel and slice the carrots into ¼" thick slices. Measure out 1 ½ quarts, then set aside.
Cut the corn from the cob and measure out a quart of kernels. Set aside.
Cut the celery into 1" slices. Measure out two cups of sliced celery and set aside.
Peel and chop the onions, then measure out two cups and set aside.
Add all ingredients to a saucepan (except the salt and pepper). Bring to a boil and reduce the heat to a simmer, simmering for 15 minutes.
Season with salt and pepper if desired.
Ladle the mixture into hot, sterilized jars, leaving 1" of headspace. Remove air bubbles, wipe the rims, then add lids and rings. Tighten until your fingertips are tight.
Load the jars into the canner and add 2" of water.
Put the lid on the canner and adjust to the locked position. Turn the heat on to medium-high and vent steam for 10 minutes. Add the gauge to the vent, if using a weighted 0 gauge canner.
Bring the pressure to 10 lbs, adjusting for altitude if needed.
Process pint jars for 55 minutes or quart jars for 1 hour and 25 minutes. Turn off the heat, then let the canner depressurize.
Remove the jars and allow them to cool for 24 hours. Label the jars and check the seals before storing.