Suppose $p>1$. For each $i\in \{1,\ldots,n\}$, consider $\varphi_i\in W^{2,p}(B_2)\cap W^{1,p}_0(B_2)$ be the solution of
$$
-\Delta \varphi_i = F_i \textrm{ in } B_2,
$$
and let $\Phi=(\varphi_i)_{1\leq i \leq n}$.

Alternatively, you could set $\varphi_i = \Gamma\star F_i$, where $\Gamma$ is the fundamental solution of the Laplacian in free space ($C|x|^{n-2}$ for $n>2$, $C\ln |x|$ when $n=2$) and $F$ is extended by zero outside $B_2$ .

In both cases, it satisfies $\| \phi \|_{W^{2,p}(B_2)} \leq C\|F\|_{L^p(B_2)}$.

Now consider $f+\textrm{div} \Phi$ on $B_{3/2}$. It is harmonic, and therefore satisfies
$$
\| f + \textrm{div} \Phi \|_{C^1(\overline{B_1})} \leq C \| f + \textrm{div} \Phi \|_{L^p(\overline{B_{3/2}})} \leq C \left( \| f \|_{L^p(B_{2})}+ \|F\|_{L^p(B_2)} \right).
$$
Finally, $$\| f \|_{W^{1,p}(B_1)} \leq C \left(\| f + \textrm{div} \Phi \|_{C^{1}(\overline{B_1})} + \| \textrm{div} \Phi \|_{W^{1,p}(B_1)}\right)\leq C \left( \| f \|_{L^p(B_{2})}+ \|F\|_{L^p(B_2)} \right).$$

The unproven fact here is the first statement, regarding the Dirichlet problem for the Laplacian. This is however true, and can be done for example using the explicit form the Green function on the ball, I believe. That doesn't hold for $p=1$ though.