To answer my own question, this follows from a more general result that is mentioned here, Theorem 4: for any convex function $\Psi: \mathbb{R}^k \to \mathbb{R}$, $$ \mathbb{E} \Psi(f(X)) \le \mathbb{E} \Psi(\frac{\pi}{2} Y \cdot Df(X)) $$ where $X$ and $Y$ are independent standard Gaussians. If you condition the right hand side on $X$ and integrate $Y$, a standard result on the moments of order-2 Gaussian chaos gives $$ \mathbb{E} (\frac{\pi}{2} Y \cdot Df(X))^p \le (cp)^{p/2} \mathbb{E} \|Df\|_F^p $$ which is what I claimed above. (By following the references a little more carefully, you can even get the sharp constant.)

To answer my own question, this follows from a more general result that is mentioned in "On measure concentration of vector valued maps" by Ledoux and Oleszkiewicz, Theorem 4: for any convex function $\Psi: \mathbb{R}^k \to \mathbb{R}$, $$ \mathbb{E} \Psi(f(X)) \le \mathbb{E} \Psi(\frac{\pi}{2} Y \cdot Df(X)) $$ where $X$ and $Y$ are independent standard Gaussians. If you condition the right hand side on $X$ and integrate $Y$, a standard result on the moments of order-2 Gaussian chaos gives $$ \mathbb{E} (\frac{\pi}{2} Y \cdot Df(X))^p \le (cp)^{p/2} \mathbb{E} \|Df\|_F^p $$ which is what I claimed above. (By following the references a little more carefully, you can even get the sharp constant.)