Since you tagged reference-request:
In the PDE/harmonic analysis literature this is a consequence of the Calderon-Zygmund Inequality, it is one of the main tools for studying elliptic regularity theory in $L^p$. It originated in Calderon and Zygmund's 1952 paper (see Chapter 3; given the age of the paper it is not surprising they state it in a way that is harder to parse using modern language, and the emphasis is on something different so the inequality you seek has to be chained together from several of the results proven). The $L^2$ version is in fact slightly earlier, due to Mikhlin (1948).
For a more modern summary, this is presented in Chapter 9 (see Corollary 9.10) in Gilbarg and Trudinger's Elliptic PDE of second order. Specifically one has the estimate that for any open bounded domain $\Omega\subseteq \mathbb{R}^n$ and any $p\in (1,\infty)$ there exists a constant $C_p$ (with $C_2= 1$) such that any $u\in W^{2,p}_0(\Omega)$ satisfies $$ \| D^2 u\|_p \leq C_p \|\Delta u \|_p $$ (here $\Delta$ is the Laplacian).