Can I use fermented milk product as starter culture for vegetable fermenting?
Yes. Whey, primarily a byproduct of cheese-making (but also from yogurt, kefir &c), can be used to inoculate a variety of ferments. In the book The Art of Fermentation, for example, there are instructions for using it as a starter for fermenting fruit, vegetable, seed/nut and sour tonic beverage ferments. The most important thing is that the whey not have been heated (most likely in cheese-making) to a point high enough to kill the bacteria present.
My experience is limited (I usually drink my whey or put it in a soup), but I'd imagine you wouldn't want to use anything milkier than whey to ferment veggies with. Fermented milk itself can be used to propagate a mother-culture (e.g. with sour cream, buttermilk, yogurt &c.) but I'm guessing you wouldn't really want to dump these into vegetables.
You can, but it depends what you are trying to achieve with your vegetable fermenting as to whether you should.
i.e., my survey of the lore of sauerkraut yielded a few suggestions to do that sort of thing, but they were definite outliers, as were the suggestions to use old sauerkraut liquid to "start" new sauerkraut, or obviously paid/pandering advice to buy starter cultures of some sort.
By far the most widely accepted view was that sauerkraut needed nothing of the sort, simply cabbage and salt and a lack of air, with the bacteria already present on the leaves doing just fine, thank you, without any "help" beyond the right amount of salt (generally accepted as 2-2.5% by weight of the cabbage) and a clean place to work, preferably without oxygen.