Let $X$ be a complex manifold. Let $E$ be a hermitian vector bundle with a given hermitian metric over $X$. On a local trivialization open subset, is there a smooth orthonormal local frame? is there a holomorphic orthonormal local frame?
Yes, there's always a (local) smooth unitary frame, just as in the real orthonormal case, as you can do Gram-Schmidt. Since the only holomorphic functions of constant magnitude are constants, the only unitary frames that can be holomorphic are constant frames on a trivial bundle.