Carbonic acid + ammonium hydroxide -----> ammonium bicarbonate + water
Balanced Equation:
H2CO3(aq) + NH4OH(aq) -----> NH4HCO3(aq) + H2O(l)
Ionic Equation:
H+(aq) + HCO3-(aq) + NH4(aq)+ + OH(aq)- -----> NH4+ + HCO3-(aq) + H2O(l)
Note: I added some space here for clarity
Net Ionic Equation:
H+(aq) + OH(aq)- -----> H2O(l)
Only the hydrogen and hydroxide ions show up in our net ionic equation because the rest are spectators. Only these two undergo a chemical reaction while the rest remain in solution.
Some interesting tidbits on ammonium bicarbonate.
When you evaporate out the water from an ammonium bicarbonate solution, you simply leave behind ammonium bicarbonate. Ammonium bicarbonate is a salt that has properties of both bicarbonate salts and ammonium salts. Consider that heating bicarbonate salts liberates loads of CO2. Likewise, heating ammonium salts liberates loads of NH3.
What's more, you liberate both loads of CO2 and NH3 when you heat ammonium bicarbonate leaving behind just the water. This is what makes it useful in baking power. Heating ammonium bicarbonate in an oven allows the baking powder to leaven breads and cakes while leaving behind nothing!