I am not well versed in cycling terminology, but assuming by BMX you mean the races such as bicycle motocross (racing on an offroad circuit, "bikros" in Czech) or bike trials (the one where your feet cannot touch the ground, "cyklotrial" or "biketrial" in Czech) and not just the various freestyle and stunt disciplines, then the answer is yes! They most certainly existed in Czechoslovakia (I imagine the rest of the Eastern Bloc might not be a particularly wild guess either), as did cross bike manufacturers.

With cycling already quite popular in socialist Czechoslovakia, it should come as no surprise that with the first BMX bikes finding their way into the country in the late seventies, various sports clubs took to the new disciplines and in a few short years regional and state competitions were organized and local bicycle factories responded by producing their own models.

Most notable by far of these was Velamos, a company whose standard bikes were also exported abroad to West Germany and the Netherlands (in addition to socialist countries), and who quickly responded to the booming popularity of BMX among Czechoslovak youth. You can see some of their fine products in this gallery. Their BMX-20 and RMX-20 models were wildly popular, although some say they were heavy, unwieldy, and sometimes painted in metallic pink which made their rider the target of playground ridicule.

In its early days BMX racing fell under the auspices of both the Czechoslovak Cycling Federation, and Svazarm, the Union for Cooperation with the Army, which although nominally tied to the armed forces was chiefly an organization promoting various sports, hobbies and activities - from programming to amateur radio to range shooting to flying planes and gliders or motorcycle racing. The last item being of most interest for obvious reasons, as motocross circuits would be appropriated for cyclist use.

The two bodies would initially set up their own bicross tracks and run their own races, with a single year-long national championship first starting in 1987. By then there were 21 registered BMX clubs in Czechoslovakia and thousands of bicycle motocross racers, with the largest events hosting easily over 500 participants.

As for trials, the story is quite similar, with small races popping up in the first part of the eighties and then growing bigger. Their popularity was helped by a feature about bike trials in the widely circulated ABC magazine – an extremely popular youth publication about all things technical, scientific and, well, generally interesting – in the summer of 1982 including a construction plan for your own trial bike (in Czech). Incidentally, the aforementioned RMX-20 model by Velamos was apparently a good candidate for a similar conversion to a trial bike.

Just for a laugh you can check out photographic evidence of a 1982 youth trial race here, or a fancy video from 1988. This will probably not be of much use to you unless you speak Czech but here is an issue of a Czechoslovak racing magazine talking about the shock local riders and bike manufacturers experienced at an international tournament held in Czechoslovakia in 1985 when confronted with western bikes which were lighter ("Their bikes weigh only 7 kilos, that's half what ours weigh!"), constructed specifically for trial racing (as opposed to being BMX racing bikes), with different tires and so on.

Perhaps you were more interested in the freestyle kind of BMX or a "BMX culture" in general in which case I am afraid I cannot help, but I hope I have at least answered some of your questions. BMX races and trials indeed existed in Czechoslovakia, and until the end of the communist regime, they would usually feature locally produced (or even home made!) bikes.