Chupacabra found its way to Belarus the same way Baba Yaga came to Mexico. Through popular culture and modern mass media.

The legend of chupacabra is actually a very recent phenomenon, as the fanciful name itself is generally attributed to Puerto Rican satirist Silverio Perez commenting on the wave of alleged sightings of the 'strange animal' and the publicized events of attacks on domestic animals sometime in late 1980s (possibly as a play on the word chupasangre or 'bloodsucker' that was used in literal sense (for e.g. mosquitoes or vampires) or metaphorically to denote exploitative people, much like its English counterpart). The attacks in Puerto Rico area were noticed since 1970s, and scientific explanations usually point to the increase of wildlife population that, unable to disperse on account of Puerto Rico being an island, began to attack domestic animals. Please note that the term literally translating to 'goat sucker' has been created after allegation of severe or complete bloodletting of the attacked animal that has been associated with chupacabra since, even though there not a single case of exsanguination has been even confirmed despite hundreds if not thousands of conducted examinations.

The rest is just a case of sensationalist journalism. The animals that were shot by the farmers and then photographed are obviously yenots (also known as raccoon dogs, manguts, tanuki or ussurian raccoons) or, specifically, its Nyctereutes procyonides ussuriensis subspecies. Although native to China, Korea and Southeastern Russia, the animal was introduced in mid-1920s to European part of USSR as a fur animal. Initially raised on the farms, raccoon dogs were gradually released into the wild since late 1920s, although the substantial increase in wild population was observed only in mid-1930s. Resiliency, limited ability to hibernate and high reproduction rate caused it to spread throughout not only USSR, but also entire Central Europe north of Balkans, Finland, Sweden and even eastern parts of France, where it became a significant invasive species. It is worth noting that the decrease in extensive industrialization caused by adoption of new technologies (and, in case of former USSR, also by the political and economic upheaval), increased environmental protection and gradual decline of hunting caused many common animals to become more and more present in the vicinity of the settlements, up to including city centres. With the packs of wild boars foraging in urban areas, frequent appearances of wolves, wild dogs, raccoon dogs or even bears in rural regions are nothing too unusual.

Now, for the main question. The allegations of 'chupacabra sightings' in Ukraine and Belarus first emerged around 2009, with the events being published in Western media around 2013. And the Ukrainians or Belarussians became familiar with chupacabras from exactly the same sources that allowed e.g. Mexicans to learn about Baba Yaga or Kaschey, that is various forms of popular media, from books and comics, to the science-fiction movies and series to books on paranormal topics. The latter were available in the communist era, and after the fall of USSR they were quickly joined by various Western media. I have quickly went over the Ukranian and Belarussian TV materials from early 2010s I found available and virtually every mention of chupacabras was made by the authors of the materials and not by the interviewed locals (with an officer of the Forestry Service explicitly stating that he 'doesn't know what kind of animal chupacabra is'), strongly implying that although known by people interested in popular culture, existence of chupacabras is definitely not a part of local folklore or even common knowledge.

So, to sum it up, the legend of chupacabra that, despite its very young age, became widely known throughout the Americas, was equally quickly transmitted abroad through various works of American popular culture.