Vicky, I spent 3 months in the Ukraine in 1995, and I was vegetarian (except for one cabbage roll, it's a long story) but not vegan. It would be hard to be vegan, if you're eating with other people or in restaurants. Things may have changed a lot, its been almost 10 years, but in 1995 all the restaurants (except one in Kiev) had only one meal, you just ate what they brought you. I ate a lot of pickles, bread, tomatoes.
Eating sour cream made things easier. Most traditional ukranian borscht was vegan, and that helped.
I found that people took my vegetarianism OK, as long as I'd eat something they made and tell them how much I liked it. It is totally different, IMO, to turn down food in a country where food security (ie, hunger and malnutrition) are real issues that people are dealing with. In Canada or the US, I don't worry about people being offended, that's thier problem. But I found it different to turn down food there, where people had way less.
Also, Ukranian sour cream is like, home made or something, it's runnier, it's really different (and delicious, I have to admit). There's no gelatin.
One last thing, your tummy may have problems with or without dairy. Bring Immodium.
And enjoy, I really loved it there.