Orthogonality of wavefunctions for different normal modes
given the above, how can one show that vibrational wavefunctions
  corresponding to different normal modes are orthogonal?


The Hamiltonian for a system with two (or more) normal modes can be written as a tensor product of (Hermitian) Hamiltonians for each normal mode. For example for two modes (stretching and bending):

$$
H_{\rm{total}} = H_{\rm{stretching}}\otimes H_{\rm{bending}}
$$

The tensor product of two Hermitian matrices is also Hermitian.
Therefore the wavefunctions are orthogonal for the same reason that you figured out on your own for the single mode case.

More generally, all Hamiltonians (Hermitian operators) have orthogonal wavefunctions (eigenfunctions) for distinct energy levels (eigenvalues). So wavefunctions corresponding to different energy levels are orthogonal as a consequence of the Schroedinger equation which is based on a Hermitian object called the Hamiltonian, and is the basis for the existence of all distinct energy states (vibrational or not) in quantum chemistry.