Being Romani myself I can only speak of Romani culture in Sweden and Spain, which are the two countries where I have family.
In my opinion, tarot does not belong to Romani people more than it belongs to any other group or community, but it is important to keep in mind that fortune-telling has played a huge part historically in Romani culture. We used to live in a matriarchy where the woman was the main bread-winner in the family, because she made money telling people´s fortune. To my knowledge though, tarot cards were not as common as regular playing cards and palm-readings, among other methods that did not require any expensive tools. The purpose was to make a living, more than practise some kind of spirituality or religion. But, as any other group that is socially and economically dependent on a more traditionally powerful group in society, we learned to read and decipher human behavior and reflect that knowledge back at the people looking to be listened to, and were willing to pay for it.
Sadly, nowadays, many Roma are embarrassed of that part of our cultural heritage and want to leave it behind, and a lot of that shame is connected to the generalization of the Roma scam artist, that of course has been created interactively with the rest of society. A huge percentage of the Roma community in Sweden now belongs to different Christian churches were fortune-telling or soul-searching via tarot-cards is banned. What we as Roma don´t realize though is that if we forget about that part of our specific history, we lose part of our cultural identity, nomatter if we want to continue practise fortune-telling or not.