Ah, a history question! When
Vancouver was established in 1824 it was under joint occupation by the U.S. and Britain, per the Anglo-American Convention of 1818. Along with the rest of the Oregon Territory, it became officially part of the U.S. in 1846 per the Oregon Treaty. It has been ever since. So, the answer is actually somewhat complicated, it was jointly occupied by the U.S. since 1824, but has been unambiguously part of the U.S since 1846.
Many years later, in 1886 the townsite of Granville in the British Dominion of Canada was incorporated to become the city of Vancouver, in British Columbia. Of course, Canada itself wouldn't become a nation for many more years.