This is an interesting question because it still has ramifications today.

The Cherokee Freedmen had an ongoing legal battle with the Cherokee Tribe based on ancestral claims of African slaves being owned by the Cherokee. In colonial times, the Cherokee were members of the dubbed Five Civilized Tribes that were viewed as more 'civilized' than other Native American groups. Part of this was in part due to embracement of more Western practices. This included ownership of African slaves. During the Trail of Tears over 1,000 African slaves owned by the Cherokee Tribe were removed along with them to Oklahoma.

After the conclusion of the Civil War, the Cherokee Nation passed the Treaty of 1866 that banned slavery unless " otherwise than in the punishment of crime, whereof the party shall have been duly convicted, in accordance with laws applicable to all the members of said tribe alike." This matched American laws at the time. The Treaty of 1866 also granted full citizenship to former slaves. These Cherokee Freedmen, sometimes of mixed African and Cherokee heritage, were granted full citizenship to the Cherokee Nation. I am unfamiliar with texts or historical works demonstrating a significant migration to Liberia, but the connection to Freedmen could prove me otherwise.

Culturally, today, the Cherokee make up the largest federally recognized Native American tribe comprised of the Eastern Band, the Cherokee Nation, and the United Keetoowah Band of Cherokee.

This all occurred before the formation of the creation of Vai, however. As for the creation of Vai ties to Liberia in terms of Cherokee, there is mixed evidence. Konrad Tuchscherer and P.E.H Hair wrote a piece about the possible connection in 2002.

They state, "no script devised in the last two thousand years is likely to have emerged totally independent of the stimulus of some diffused knowledge of previous scripts..." "For over one hundred and fifty years students of the history of writing and the development of scripts have been intrigued by the modern inventions of the Cherokee script, devised in 1821, and the Vai script, devised 1832/33, these being the earliest indigenous script inventions in North America and sub-Sarah Africa respectively (and each continuing in use until today, thus arousing further interest and commitment, scholarly and otherwise). The two scripts have often been cited in parallel as examples of so called "independent inventions among natives." True, the chronology of the inventions is curiously quasi-coincidental, the two dates being little more than a decade apart, while scripts share the same orthographic organization, both being syllabaries. But the similarities end there. The outward forms of characters share almost no graphic similarity."

The author alludes to a possible connection between North American Protestant missionaries that were involved in works with both the Cherokee and the Vai peoples of Africa and a dissertation written by Svend Holsoe that mentions a Cherokee man that immigrated to Liberia in 1823 named "Agustus (or Augustin) Curtis" and that he had possibly settled "at Cape Mount in Vai country as early as 1829." The authors state that Holsoe "was careful to point out that there was no direct evidence that Curtis was an intermediary." The potential may exist for a connection, but the hypothesis has yet to be confirmed, AFAIK.

Sources:

Tuchscherer, Konrad, and P. E. H. Hair. "Cherokee and West Africa: Examining the Origins of the Vai Script." History in Africa29 (2002): 427-86. doi:10.2307/3172173.

http://www.vcdh.virginia.edu/liberia/index.php?page=Stories&section=Augustus+Curtis