Final answer:
In neither Hans Christian Andersen's original fairy tale nor the Disney adaptation of "The Little Mermaid" do Ariel's sisters make a vow about what they will do until Ariel returns.
Step-by-step explanation:
In the story of "The Little Mermaid," the answer to what Ariel and her sisters vow to do until Ariel is home is not distinctly provided in the classic version by Hans Christian Andersen. In the Disney adaptation, which is more familiar to many, there is a strong sense of family, but the specifics of such a vow are not a central plot point and the movie does not explicitly state that Ariel's sisters made a vow about her return. If this concerns a particular adaptation or retelling that includes a specific vow, providing the specific source or version of "The Little Mermaid" in question would be essential for an accurate main answer. If we stick to the most renowned versions, neither the original fairy tale nor the Disney movie includes a scene where Ariel's sisters make such a promise.