It is possible to preserve vegetables with salt via lactic fermentation but as GdD mentioned, 1 cup is way too much. I believe lactic fermentation is your best bet however, either dry salting or brining. If you're dry salting the maximum you'd use is 20 - 25% by weight but the vegetables would require several rinsings to get the saltiness down to a tolerable level. At the low (and more common) end, where you're brining, you'd use about 2 - 5% salt by weight. Just don't go below 2% or you risk ruining your batch by inviting other stuff in.
I would look at a few recipes for lacto-fermented tomato sauce and modify one to suit your preference. Just remember that fermentation will continue for as long as you let it. Don't use a tight-fitting lid or gas buildup will create an explosion hazard. The flavor will also continue to get stronger over time. Storing it in your fridge or a cool place will slow the fermentation enough to give you several months with it tasting more or less the same. I've heard of people pasteurizing their ferments in jars to extend the shelf life further but have no personal experience with it.