Does the ammonium ion hydrogen bond with water? 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.
There is a detailed consideration of this issue in  Hydrogen Bonding of the Ammonium Ion Journal of Molecular Structure 345 (1995) 77-81.

First, the try to be clear about what they mean by "hydrogen bond":


  For the purposes of the discussion, a hydrogen
  bond is taken to be a specific short-range interaction
  in which the N-H bonds within the
  ammonium ion are significantly affected by the
  surrounding ions, and not simply electrostatic
  attraction.


They conclude that ammonium ions rotate too freely in water for significant hydrogen bonding to exist.  

In gas phase calculations, they find that one water molecule hydrogen bonds to one ammonium ion (in other words, the dimer is hydrogen bonded), based upon the lengthening of the N-H covalent bond, but as you add 2, 3, and 4 water molecules, the N-H bond length returns to the free ammonium N-H bond length and the O --- H length increases.