Here's a simple and far-from-optimal condition guaranteeing uniform convergence.

Suppose for that there exists $C>0$ such that for any positive integer $n$ $\newcommand{\bR}{{\mathbb{R}}}$

$$
\int_\bR |f^{(n)}(x)| e^{-x^2/2} dx\leq C^n.
$$

In this case the associated Hermite series  is

$$
\sum_{n\geq 0} \frac{c_n}{n!} H_n(x),
$$

where

$$
c_n=\frac{1}{\sqrt{2\pi}}\int_\bR f^{(n)}(x) e^{-x^2/2} dx.
$$

and this converges to $f$ uniformly on compacts. This follows [from known asymptotic estimates for Hermite polynomials][1].

For more precise results, you need to look at Gaussian-Sobolev spaces and the Ornstein-Uhlenbeck operator $H$.

  [1]: https://dlmf.nist.gov/18.15#v