How do these solutions "react"?
You picked a difficult question because this question involves acid-base reactions. It is no longer a simple "double displacement" reaction where the cations and anions "swap partners".


  If I were to combine solutions of $\ce{(NH4)2CO3}$ and $\ce{HNO3}$ in water, the only ionic compounds that could form are $\ce{H2CO3}$ and $\ce{NH4NO3}$ right?


No, not really. You have to realise that the protonation of $\ce{CO3^2-}$ occurs in two stages: firstly, to $\ce{HCO3-}$, and then only to $\ce{H2CO3}$. Also, the $\ce{NH4+}$ ion is acidic, and can react with water to form $\ce{NH3}$.

On top of that, $\ce{H2CO3}$ will spontaneously decompose into $\ce{H2O}$ and $\ce{CO2}$, the latter of which is released as a gas.

Since these species are all soluble in water, and soluble ionic compounds exist as ions in water, a more appropriate way to think about this question is "what species are present", not "what ionic compounds are present". Species is just a general term that includes ions as well as molecules (since, for example, $\ce{CO2}$ and $\ce{NH3}$ are not "ions").


  But since both are soluble no precipitate would form correct?


That is correct.


  So does that mean that no reaction occurred?


No, there is a reaction happening because of the $\ce{CO2}$ gas formed. You will observe a lot of effervescence if you were to mix these two together.

However, if this question did not involve $\ce{CO2}$... Let's say you are reacting $\ce{NaCl}$ with $\ce{KBr}$. Since all the possible products ($\ce{NaBr}$ and $\ce{KCl}$) are soluble, you could say that in this case, no reaction would occur.


  Does it also mean that an ion like $\ce{NO3-}$ will never react in an aqueous solution because it is always soluble in water?


No. You could say it will never be precipitated out. However, $\ce{NO3-}$ can also react in other ways apart from precipitation. For example, it can be used as an oxidising agent. One famous example is the dissolution of a copper coin in concentrated nitric acid.