Yes, but very late -- after censorship finally drew down a bit in the 1980s.

Horror movies worldwide go back quite early, with even the the Lumière brothers making an initial effort, and the three-minute Le Manoir du Diable from Melies. Before the Soviet Union there were Russian movies like The Terrible Vengeance and Satan Triumphant.

Once the Soviet Union proper started (and up to the aforementioned 80s), it becomes much harder to find examples, and there's something I'll call the "folktale exemption" that has to be activated. Before I get into some actual movies, let me explain why horror movies were so rare.

Soviet censorship was very heavy. In the Lenin-Stalin era movies leaned to pure propaganda, and even after the death of Stalin it was very much not suggestible there were any flaws in Communism. Quoting the filmmaker Andrei Smirnov, most famous for his work in the 70s:

There were no clearly formulated rules. Everything depended on the particular official who said yes or no. The Soviet State Committee for Cinematography was just the final stage; censorship started on the first day of film production. At Mosfilm studios, local editors proofread the scripts, after which they were discussed by the arts council – a team of filmmakers, screenwriters and directors. And this wasn’t done without the Party Committee, the local body of the Communist Party. And of course, they closely observed the films during filming.

Many movies simply never saw the light of day. It was much easier for film-makers to return to easy touchstones than to risk ostracization. Even when a film came out, if its politics or events fell on the wrong side they could be suppressed; The Cranes Are Flying from 1957 did well at the Soviet box office but Khrushchev disliked it (a soldier's girlfriend during WW2 is "unfaithful") so Mikhail Kalatozov's name was not even mentioned in the newspapers when the film won the Palme d'Or at Cannes.

For horror in particular, there were a couple elements in Soviet censorship that stacked the deck against it. First, sex and extreme violence were out, both definite staples of the genre. Additionally, superstition was looked down upon, and the idea of suggesting demons and the like were somehow "real" was a holdover of backward unenlightened times. Class struggle was meant to be the defining feature; there could be no irrationality or self-reflection on the internal horrors of society.

Having said all that, a few works that could be considered horror snuck through. The Bear's Wedding from 1925 involves a countess in the 19th century who is scared by a bear when her child is born, giving the child when he grows up a desire to dress up in bearskin and murder women, including his new wife. (You can watch the entire film at the link here.) It gives the impression more of a cautionary fairy tale than straight horror.

The Night Before Christmas (1961) involves a blacksmith Vakula who makes a deal with the Devil, which sounds like a possible premise for horror, but Vakula manages to successfully trick the Devil; despite leaning toward some appropriate imagery it is really a straight fantasy. (Here's a clip to show what I mean.)

There is one "true" horror movie from around this period; sometimes it gets called the only horror movie made by the Soviet Union (by people who I presume haven't watched any late-Soviet movies, but for this period maybe the assessment is fair). That would be Viy from 1967.

The movie's horror flair arguably was mostly the fault of Alexander Ptushko, who generally worked with Russian fairy tales, carefully avoided the horror, and made skillful use of stop-motion animation. His movie Sadko in particular was rearranged as The Magic Voyage of Sinbad for American audiences (although it's originally based on a Russian story). He did the special effects for Viy. The two directors, Kropachyov and Ershov, were formerly in production design; this was their first stab at direction. Despite prior fairy-tale edits around horror, this particular story -- based on a Gogol story -- drives right through it. A priest murders a witch and has to stand vigil over the body for three days in a protective circle of chalk. The nights amp up considerably in terrors, as you can view in the trailer. (There's also a 2014 remake in 3D.)

The movie decidedly stands out amongst others made at the time, and it's only the continuation of the fairy-tale notion that even allowed it to be made in the first place. Gogol claimed the Viy itself -- picture here -- is from folklore, but it is generally considered to be the author's own invention. This allowed the movie to slip under the fairy-tale exemption, however, being not "real" horror despite it definitely containing all the elements.

Jump up to the 1980s -- Gorbachev, glasnost, and all that, and some opening up of media permissiveness even before it became official -- and you definitely started to get horror. So much that it'd be possible to go book length (in fact, Fear Before the Fall: Horror Films in the Late Soviet Union is upcoming for January 2023). But let me just sample a few movies to give you an idea where horror started to tend when it opened up:

Savage Hunt of King Stakh (1980): A researcher goes to investigate folk legends. (Still one foot in that tradition!)

Mister Designer (1987): Artist tries to gain eternal life through the use of mannequins.

Dogs (1989): Hunters go after killer dogs: killer dogs go after hunters.

Daddy, Father Frost is Dead (1991): an "underground" movie from the "necrorealism" genre; the term is a parody of Socialist realism; let me just quote the blurb from sovietmoviesonline, where you can watch the film as well:

A biologist, obsessed with the idea of writing a treatise on a new kind of mouse, becomes witness to a number of bizarre and horrific events, from his son’s suicide, to the S&M engaged in by respectable middle-aged men, to his own family’s psychic morbidity.

As far as what manifested, in an analytic sense I'd say maintaining a quarter of the fairy tale sensibility while also going very, very, dark. I think perhaps the most succinct I can put it is to summarize a post-Soviet movie which feels like it comes from the same trend: Silver Heads from 1998. Scientists do an experiment to merge man with tree. This is of course a very logical thing to do: man will become much more sturdy in the process! Researchers go in the forest to do the experiment, and they are not alone, as there are other creatures from the past with them.

Notice the merging of a.) odd, bureaucratic logic which leads to absolute horror b.) belief in the power of science c.) fairy tale fringes with the creatures in the forest d.) expressionist, confusing moments and e.) a completely black premise.

You can watch the entirety of the movie here, if you dare.

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Rollberg, P. (2016). Historical Dictionary of Russian and Soviet Cinema. United States: Rowman & Littlefield Publishers.

Paszylk, B. (2009). The Pleasure and Pain of Cult Horror Films: An Historical Survey. United Kingdom: McFarland, Incorporated, Publishers.