It says on the steam page system requirements that the default API is Vulcan 1.1. Does that mean I need that specific version to run BG3 or can I do that with the latest version of Vulcan.
You should not need to be concerned any of that. All needed parts will come with installation of Baldur's Gate 3 game and installation of nvidia drivers.
P.S. In technical perspective vulkan uses backwards compatibility. 1.3 vulkan is backwards compatible to 1.1 software.