There is such a thing as vegetarian pepperoni, although I've never tasted it. (I prefer turkey pepperoni myself.)
It's put out by "Smart Deli", & is not only vegetarian, but vegan as well. Most major supermarkets carry some of the "Smart Deli" products (there are quite a few), normally found in the produce section or wherever they have their refrigerated tofu & such. Your best bet may be a place like Whole Foods/Fresh Fields, or some other upscale place that caters to natural foods, although our Wal-Mart here does carry a number of Smart Deli products as well. You'll probably just have to search around.
Do keep in mind that there are many perfectly authentic variations of borsch that naturally contain no meat at all. Perhaps you'd be better off researching some of those.
(P.S. My husband is Ukrainian, & he's absolutely positively hated borsch - meatless or not - his entire life, especially as a child. Maybe you should think about some other recipe