Indeed, as @TomAu has said, the evacuation of the Czechs (and Slovaks!) was handled by the US government, which also presumably paid for it.
I found a Romanian article about Victor Cădere, a Romanian diplomat operating at the same time and place with a similar objective: evacuation of Romanian nationals. This article provides some valuable information:
Meanwhile, the Allied Forces convened in Paris and decided that they
first assess the means and the ways to meet the objective, i.e.
prisoners' repatriation. The Convention between the United States and
Great Britain, held at the end of 1919, decided that by collective
effort of the two states some 64,000 people be repatriated. The United
States were assigned the mission to deal with the Czechoslovak troops,
i.e. some 35,000 people, whereas the British undertook to board the
other troops, i.e some 28,500 people. According to a financial
agreement concluded, Great Britain was to raise the amounts necessary
to transport the people from the respective governments whose troops
were fighting in Siberia. However, the United States benefited from
different conditions – the amount of $12 million was lent to the
Czechoslovak Government to ensure the transport of troops.
I don't quite understand the financial arrangements as detailed there, though: who was supposed to lend $12 million to the Czechoslovak Government?
Also, two ships that took part in the evacuation are identified there:
Indeed, wikipedia tells us about the former ship:
President Grant was transferred to the Army 6 October 1919. During the
period January to November 1920, she served as the United States Army
Transport Republic and made two voyages repatriating Czechoslovak
troops from Siberia to Trieste.
Another ship I was able to find in wiki is USS Mount Vernon.
Finally, a master's thesis by Major Robert Dziak of the Czech army tells us that:
Totally, 36 transports were dispatched and over 67,700 people were
transported through three main directions: First, around Asia, then
via Suez Canal to Trieste; Second, across Pacific ocean, Panama canal,
Atlantic ocean to Trieste or Hamburg; Third, through Pacific ocean to
Canadian West coast, by rail road across Canada, then through Atlantic
ocean to Hamburg. Many passengers were not members of the Legion. The
transports took also care of the Legionaries’ family members, civilian
personnel, and POWs.
