For any language in $\mathsf{NP}$ there exists a proof that can be verified using $O(\log n)$ working space. One just needs to use the same ideas used to prove SAT is $\mathsf{NP}$-complete. By definition, given an $\mathsf{NP}$ language $L$, we know that there exists a turing machine $M$ such that for any $x \in L$ there exists a $y$ such that $M(x, y)$ accepts. We can construct a logspace verifiable proof for $x$ by writing down $y$ and the computation tableau of $M$ on input $x, y$. It is easy to verify in logspace that the tableau describes a valid accepting computation of $M$. Similarly, for any $x \not \in L$ and any $y$, no valid computation of $M(x, y)$ accepts, so the logspace verifier won't accept any tableau.
Of course this does not show that $\mathsf{NP} = \mathsf{NL}$ (because that would imply $\mathsf{NP} = \mathsf{P}$). The reason is that the verifier has two-way access to the proof (can go back and forth). The proof-verifier definition of $\mathsf{NL}$ gives the the logspace verifier only one-way access to the proof (once a bit of the proof is read and the head moves right it cannot move left).