... does the Supreme Court interpret “race” to include ethnicity in this context?
Fifteenth Amendment Right to Vote Clause: Doctrine and Practice
Although the "immediate concern of the Amendment was to guarantee to the emancipated slaves the right to vote," the Amendment "is cast in fundamental terms, terms transcending the particular controversy," and "grants protection to all persons, not just members of a particular race." Moreover, the Court has construed "race" broadly to comprehend classifications based on ancestry as well as those based on race. "Ancestry can be a proxy for race," the Court has explained, finding such a proxy in Hawaii’s limitation of the right to vote in a statewide election for an office responsible for administering a trust for the benefit of persons who can trace their ancestry to Hawaiian inhabitants of 1778.
Ethnicity was first used in 1920. The Court's decision, in Guinn v. United States, 238 U.S. 347, referring to ancestry was given in 1915. However, ancestry is considered a component of ethnicity.
A later Fifteenth Amendment case, Rice v. Cayetano, 528 U.S. 495, 514, made indirect references to ethnicity by using the adjective ethnic. In Rice,
The State's ancestral inquiry is forbidden by the Fifteenth Amendment for the further reason that using racial classifications is corruptive of the whole legal order democratic elections seek to preserve. The law itself may not become the instrument for generating the prejudice and hostility all too often directed against persons whose particular ancestry is disclosed by their ethnic characteristics and cultural traditions. The State's electoral restriction enacts a race-based voting qualification.