The answer is no for irreducible $f(x) \in \mathbb{Z}[x]$ of degree at least $3$. In this case, let $F(X,Y)$ be the homogenization of $f$, so that $f(x) = F(x,1)$. Let $\{p_1, \cdots, p_k\}$ be a fixed set of distinct rational primes. Then it was shown by Mahler and many subsequent authors that the equation $$\displaystyle F(x,y) = p_1^{z_1} \cdots p_k^{z_k}$$ where $z_i$ range over all integers and $x,y$ are integers, has at most $$\displaystyle (5 \times 10^6 \deg f)^k$$ solutions; see

J.-H. Evertse, "The number of solutions of the Thue–Mahler equation" J. Reine Angew. Math. , 482 (1997) pp. 121–149.

Therefore, this also gives an upper bound for the number of solutions to the equation $f(x) = n$, where $n$ is a number all of whose prime divisors come from some finite set.

The answer is no for irreducible $f(x) \in \mathbb{Z}[x]$ of degree at least $3$. In this case, let $F(X,Y)$ be the homogenization of $f$, so that $f(x) = F(x,1)$. Let $\{p_1, \cdots, p_k\}$ be a fixed set of distinct rational primes. Then it was shown by Mahler and many subsequent authors that the equation $$\displaystyle F(x,y) = p_1^{z_1} \cdots p_k^{z_k}$$ where $z_i$ range over all integers and $x,y$ are integers, has at most finitely many solutions, with the record $$\displaystyle (5 \times 10^6 \deg f)^k$$ being obtained by Evertse; see

J.-H. Evertse, "The number of solutions of the Thue–Mahler equation" J. Reine Angew. Math. , 482 (1997) pp. 121–149.

Therefore, this also gives an upper bound for the number of solutions to the equation $f(x) = n$, where $n$ is a number all of whose prime divisors come from some finite set.