Vault-Tec was US-Only, and for good reason
Fallout Brotherhood of Steel (the game) states that Vault-Tec was so intertwined with the war departments doing research for the USA that they were pretty much a department of the government by the time the bombs fell.
BOS content is not canon, but is allowed as "flavor material" for the stories.. I rate the above nugget of info as "flavor" and therefore acceptable.
Project Safehouse and FEV were not something the USA would want to leak out
Project Safehouse was the Enclave's dastardly plan to use vaults as social experiments on unwilling and uninformed human subjects, specifically because the Enclave believed that nuclear war was inevitable; Project Safehouse was to provide them with research allowing them to survive this inevitable future. The Enclave wouldn't have allowed any of their research to leak out of the USA, even if there was a market for it before the Great War. They would make Vault-Tec rich, but in exchange Vault-Tec would pretty much end up under the control of the Enclave.
FEV was a project of West-Tec, not Vault-Tec. However, Vault-Tec was at least aware of Forced Evolutionary Virus as at least one vault (Vault 87) was specifically designed to do research on FEV. Leaked data about FEV was part of the final trigger to war; the USA would have done anything to keep data about their biological weapon under wraps.
These two data points point toward the USA being the only (legal) market for Vault-Tec.
There probably wasn't a market anyway.
The Resource Wars and the nuclear exchanges in the Middle East had pretty much destroyed Europe and the East; governments were shattered and the civilians were struggling for survival. China was in full throated competition with the USA at this time; Vault-Tec would NOT have contracted with them. Only the USSR would have been a viable market for Vault-Tec.
Canon is pretty quiet about the Soviet Union, but there is some data available. There was a consulate between the USA and USSR at the time of the Great War, so diplomatic relations were good. People freely travelled between the two countries. The Soviets were not in direct conflict with the USA during the Resource Wars, implying they either had their own supplies of energy (nuclear, perhaps, or more likely natural gas) or otherwise found a way to survive without fighting. There's a strong hint in canon that China was essentially committing "suicide" against the USA; knowing that oil was almost gone and neither side really having any alternatives, China decided to do an all-out conventional war for Alaska, a really rediculous plan from a strategic point of view, as the army would consume more resources than they could gather.
The USSR would want to stay out of the way, as they would need to husband their resources carefully during the war, as opposed to China's wasteful strategy. It's likely the USA would not want Vault-Tec to work with the USSR in fear that anything might leak to China, or in fear of their only remaining "semi-ally" becoming fully aware of FEV, or America's Fusion revolution.
If you accept that China was committing "suicide by cop", the USA's revelation of Fusion Batteries as a backup power source required the firing of nuclear weapons. Fusion Power would give the USA infinite power, while China had spent it's last drops in the war. The USSR would likely become an enemy as well at this point, seeing Fusion Power as a destabilizing element if both sides didn't have it.
There's some speculation above, of course, as nothing much is said in canon for the USSR, but it's pretty solid that the USA kept Vault-Tec to itself.