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 $\Phi=(\varphi_i)_{1\leq i \leq n}$. It satisfies $\| \phi \|_{W^{2,p}(B_2)} \leq \|F\|_{L^p(B_2)}$. 

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.