I am afraid that troika rides are not easily available in Ukraine. Online search on troika rides in Ukraine gives no meaningful results. I think I can explain why.
Troika is a cultural icon of Russia, not Ukraine. And seeing how are the political relationship between Russia and Ukraine - nearly open hostilities - I would not expect to find troika rides available.
Ukraine was once a part of Russia/USSR and there are tendencies recently to eliminate all cultural symbols related to the Russian/Soviet times. While troika is not an exemplary symbol in such respect, it is a symbol of Russia nevertheless. So, I would expect it to be difficult to operate troika rides from the current cultural and political standpoints.
Nearest place to Ukraine to ride troika would, in my opinion, be Russia. Multiple equestrian clubs are offering "an authentic Russian experience of riding Troika", such as: 1 2 3 4 and many others (in Russian). English offering for foreign tourists are e.g. here