Can I dissolve eggshells in vinegar, and use the vinegar in a dish? [closed]

Addressing the food chemistry aspect:

Vinegar (acetic acid) reacts with calcium carbonate in the eggshells to make calcium acetate (Wikipedia), as in the naked egg experiment. Calcium acetate can be used, among other things, to gel alcohol; in food it can be used to coagulate tofu as well as having a stabilising effect. There's a chance it will act as a thickener depending on quantity depending on what else is present in the sauce; the chemistry is quite intersting.

This chemical reaction means you won't have as much acidity left in the vinegar as when you started but the additional flavour of all but the most refined vinegars should survive, so you can choose one that matches the other ingredients. For a Chinese-inspired dish, rice vinegar would be an obvious choice. It's not clear to me whether the calcium acetate itself would have a flavour, pleasant or otherwise

I can't comment on the nutritional aspect, but will just add a note of caution from a food safety point of view (actually 2):

Calcium acetate has a medicinal use and making and consuming medicinal compounds at home can't be assumed to be safe. In this case it treats cases of excess blood phosphate by preventing the absorption of phosphate in food; how much you'd consume, and how this relates to the levels seen in foods/medicines are not things we can help with.

If you wished to make a vinegar-preserved food (rather different to the example you've added since I started writing this) you'd have to take into account the fact that you've neutralised much of the acid in the vinegar, and therefore much of it's preserving power.