This is pedantic and trivial query.
For this below reaction,
$$\ce{NH4Cl + NaOH -> NH3 + H2O + NaCl}$$
we can reduce it to its net ionic equation
$$\ce{NH4+ + OH- -> NH3 + H2O}$$
Is it correct to say that $\ce{NH4Cl}$ is the acid or that $\ce{NH4+}$ is the acid?
I also wonder if the $\ce{Cl-}$ ions have any direct impact on the acid-base proton transfer in this reaction? Then at least one can conclude if $\ce{Cl-}$ is absolutely only a spectator ion.
I think since your question is a symantic/pedantic one (I dont say that as any sort of insult) we'd need to consider the definition of an acid. I think IUPAC would be considered the definitive source.
They define acids as: A molecular entity or chemical species capable of donating a hydron (proton) (see Brønsted acid) or capable of forming a covalent bond with an electron pair (see Lewis acid).[link]
I think you could reasonably argue that NH4+ is a "molecular entity" while NH4Cl is a "chemical species", both of which are capable of acting as Bronsted acids. I think the answer is both.
Those saying that NH4Cl isnt an acid - would you also not consider HCl to be an acid? This seems a slightly bizarre statement tbh.
Ammonium ion $\ce{(NH4^+)}$ is the acid part. Ammonium chloride $\ce{(NH4Cl)}$ itself is an acidic salt.
And chloride or any other "spectator" ion does have an impact on the reaction, as spectator ion's affinity towards the acidic part, and also spectator ion's stability by itself in the solution, will determine the solubility/acidity of the entire compound (here, ammonium chloride). Even though this is not apparent from the net ionic reduction, it is non-negligible.
Firstly the second reaction is reversible. Secondly, The chloride ions are spectator ions in this reaction. They are not involved between $\ce{NH4+}$ and $\ce{OH-}$. They remain in solution as chloride ions throughout the reaction. $\ce{NH4+}$ behaves as a Brønsted-Lowry acid (proton donor), as it donates a proton in the reaction. Since $\ce{NH3}$ is protonated in the reaction, it functions as a base. Since $\ce{NH4+}$ functions as the Brønsted-Lowry acid in the opposite reaction, it is known as the conjugate acid of $\ce{NH3}$.
You can surely conclude the acidic nature of $\text{NH}_4^+$ using the ionic reaction mentioned.
Otherwise, you can also conclude the same by acid-base the reaction through which it is formed: $$\text{NH}_3 + \text{HCl} \longrightarrow \text{NH}_4^+\text{Cl}^-$$ Here, $\text{NH}_3$ is a comparatively weaker, Lewis base (since it has a donatable lone pair) while $\text{HCl}$ is a strong protonic acid. Thus, the salt formed by this acid-base reaction will show a mild acidic character.
The above reaction can also be represented as the following ionic reaction: $$\text{NH}_3 + \text{H}^+ \longrightarrow \text{NH}_4^+$$
Now, note that $\text{NH}_4^+\text{Cl}^-$ can be considered equivalent to $\text{NH}_4^+$ in most of the reactions since $\text{NH}_4\text{Cl}^-$ dissociates into $\text{NH}_4^+$ and $\text{Cl}^-$ in the aqueous solution and thus $\text{Cl}^-$ has no direct impact on the reactivity of $\text{NH}_4^+$. The only thing keeping them together in a non-aqueous solution is the ionic attraction between them.
Now for the final question:
Is it correct to say that $\ce{NH4Cl}$ is the acid or that $\ce{NH4+}$ is the acid?
The thing is, theoretically when we are talking about acid-base reactions, $\text{NH}_4^+$ is considered as the salt product of acid-base reaction of ammonia and hydrochloric acid, which are far more reactive species and stronger base and acid, respectively. Hence we actually classify it as a conjugate acid, that is the product formed due to protonation of a base. Here, the lone pair of Nitrogen atom of $\text{NH}_4^+$ are providing with the electron pair to the proton ($\text H^+$), giving the nitrogen atom a positive charge. It is the conjugate acid of the base, ammonia.
Please note that the term conjugate acid or conjugate base are reaction specific. That is, these are used only for the product of an acid-base reaction. For example, I already explained that $\text{NH}_4^+\text{Cl}^-$ is the conjugate acid of ammonia. However, if it further reacts with a base, say: $$\text{NH}_4^+ + \text{OH}^- \longrightarrow \text{NH}_3 + \text{H}_2\text{O}$$ Since $\text{NH}_4^+$ has donated an electron here, it behaves as an acid and ammonia is its product, hence ammonia is the conjugate base of $\text{NH}_4^+$ while $\text{H}_2\text{O}$ is the product due to acceptance of a proton by $\text{OH}^-$ , so $\text{H}_2\text{O}$ is the conjugate acid of hydroxide ion.
Conjugate acid of a strong base is a weak acid, meanwhile conjugate base of a strong acid is a weak base. Conversely, conjugate acid of a weak base is a strong acid and conjugate base of a weak acid is a strong base.
So concluding the answer here, $\text{NH}_4^+\text{Cl}^-$ is the ionic salt while $\text{NH}_4^+$ is a weak acid or conjugate acid of ammonia.
