It depends on the Native American nation in question. This photo is of a leader of the Taku Tlingit in Southeast Alaska. As he was born in the mid-19th century in an area without European settlement, he was presumably a “full blood,” and that thing on his face appears to be a beard. Younger men at the time were known to pluck their facial hair, but it seems they could indeed grow it.

As in the rest of the world, I assume some groups and individuals were more hairy than others, but it certainly isn't true that all Native Americans lacked the ability to grow facial hair.