How do these hold up to canning? Could I just throw these jars in a water bath after and have some shelf stable veggies? If so, how long in the bath?

Well, you can preserve giardiniera in a water bath canner- but not one using this recipe, unfortunately. You can't safely can any sort of oil, at least not in those amounts. BUT if you like this recipe, you could use it with a few changes. Omit the oil, and add in 2 cups of water instead. Heat the brine and spices to boiling, then drop in your chosen veggies. Bring that back to a boil, and turn it off. Divide your hot veggies between your pint jars, then add in the hot brine and stop to leave a 1/2 inch of head space between the veggies and your sealing lid. Then water bath can or steam can those for 10 minutes for a pint, 20 for a quart jar. When you open a jar, you can drain half the brine and add in that amount of oil- and keep it in the fridge. That's the only way to safely have it shelf stable. Otherwise, you can follow the recipe as written, but it will have to stay in the fridge. Instead of having six individual jars though, you could keep it in one large gallon container too. I make my own giardiniera as well, the shelf stable variation though, and this is how I do it. You can add in the oil later, without any problem. It doesn't really add in anything in the sitting, it just balances out the brine. So you can add it after opening easily enough- and have the best of both worlds.