Given the reaction :ammonia gas and carbon dioxide gas are bubbled into water or
NH3 (g) + CO2 (g) + H2O (l), I do not understand how the products become (NH4)2CO3. Instead, I thought the product would be (NH4)HCO3 because the NH3 would take an H from water to become NH4+ and the CO2 would take the remaining OH to become HCO3-.

You will get (NH4)HCO3 only when CO2 gas is passed in excess.