If you are interested in $L^p$-theory, you are probably looking for solutions belonging to a Sobolev class $H^{s,p}(\Omega)$ with some $s>0$ and $p>1$. In this case, the Besov space
$B^{s-1-1/p,p}(\partial\Omega)$ is the "right" trace space. In particular, the restriction map
$$\rho: H^{s,p}(\Omega)\to B^{s-1-1/p,p}(\partial\Omega)$$
$$u\mapsto \partial_{\nu} u$$
is well defined for all $s>1+1/p$ and is surjective.

As for the Neumann problem, the following result is true

**Theorem.** Let $s>1+1/p$ where $1< p< \infty$. Then the Neumann problem
$$\begin{cases} \triangle u=f & \mbox{in }\Omega,\\\ \partial_{\nu} u=\phi & \mbox{on }\partial\Omega\end{cases}$$
has a unique solution $u$ in the space $H^{s,p}(\Omega)$ for any $f\in H^{s-2,p}(\Omega)$ and any $\phi\in B^{s-1-1/p,p}(\Omega)$.

Have a look at the very accessible exposition by Kazuaki Taira.