Gods can indeed be of a specific race, especially because some gods were previously mortals. The Faerunian Pantheon is the groups of gods that people know the best, which is mostly human gods, but also mostly gods of the Sword Coast. Other areas of Faerun have their own groups of gods, such as Mulhorand, which has their own pantheon of egyptian-flavoured gods.
But we also have gods separated into racial pantheons, such as the Seldarine, or Elven pantheon. Whether a racial deity accepts worship from other races depends entirely on how xenophobic said gods are, which usually corresponds to how xenophobic the specific race tends to be. Most races have their own deities, though the size of their pantheons varies wildly, and can be as small as one racial god.
But gods are also very changeable, at least in the realms, which is the root cause of a lot of confusion about what gods are alive or dead. Bahamut is a dragon god, specifically a god of dragons, but began to accept human worshippers after a lot of shenanigans with being demoted and being made subservient to Torm, and then being promoted by Torm, and is the only god in the draconic pantheon to do so.
Here is a page of pantheons on the realms wiki, but even this is not 100% comprehensive, as it's missing the lizardfolk pantheon I think.
