It is clear that all constant polynomials work. Assume $\text{deg}f \geq 1$. WLOG, we may assume that the leading coefficient of $f$ is positive. (Otherwise, just work with $-f$.) Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ where $a_i \in \mathbb{Z}$. Now we can choose sufficiently large $m \in \mathbb{N}$ so that all the coefficients of $f(x+m)$ are non-negative. If we set $g(x) = f(x+m)$, then it is evident that all the numbers $g(10^k)$ for all sufficiently large $k$ have the same digit sum.