Short Answer?
There isn't any room.
Nitrogen can only bond with 4 other atoms.
Normally it's three, but a proton is so strongly charged that it forms a dative bond with the Nitrogen.
My understanding is that a dative bond is indistinguishable from an ordinary covalent bond, the only difference is that both electrons come from the same atom (a lewis base) so the distinction is completely artificial.
Each Hydrogen atom can only bond to one other atom (ignoring boron chemistry, and hydrogen bonds), and they're all bound to the central Nitrogen.
So the Ammonium (NH4)^+ is incapable of forming any further covalent bonds, it does, however, have an over all positive charge, like a metal ion, so it's capable of attracting an ion with a negative charge and forming an Ionic bond - which is simply an electrostatic attraction between two charged particles (ignoring certain complicating things like hardness and softness).
Now, I admit, that things aren't quite as clear cut as I have made them out, but it makes the explanation clearer.
I hope it helps.