What about oxonium ions, such as hydronium ion? Do these hydrogen bond with water?
If we see hydrogen bonding as a purely electrostatic phenomenon, then why not? Ammonium ion still has nitrogen bearing a partial negative charge. Of course it can interact with positively polarized hydrogens in an electrostatic way.
However, we also know that hydrogen bonding is partially covalent. So, it seems unlikely that the ammonium ion would be able to form any additional covalent bonds.