I found an answer to my question (well almost, I'd still like a more explicit description) in the following paper:

MR894515 (88k:13004) 13B05 (11T99 20J06) Landweber, Peter S. (1-RTG); Stong, Robert E. (1-VA) The depth of rings of invariants over finite fields. Number theory (New York, 1984–1985), 259–274, Lecture Notes in Math., 1240, Springer, Berlin, 1987.

In it they prove (their Theorem 5) that if $V$ is a finite dimensional vector space over a field $k$ of characteristic $p$, and $G$ is a finite group acting linearly on $V$ that if $V_G$ the subspace of covariants $\{ gv - v : v \in V \}$ has codimension 1 in $V$ (which is the case for my question), then the ring of invariants $k[V]^{G}$ is a polynomial ring.

I found an answer to my question (well almost, I'd still like a more explicit description) in the following paper:

MR894515 (88k:13004) 13B05 (11T99 20J06) Landweber, Peter S. (1-RTG); Stong, Robert E. (1-VA) The depth of rings of invariants over finite fields. Number theory (New York, 1984–1985), 259–274, Lecture Notes in Math., 1240, Springer, Berlin, 1987.

In it they prove (their Theorem 5) that if $V$ is a finite dimensional vector space over a field $k$ of characteristic $p$, and $G$ is a finite group acting linearly on $V$ that if the subspace of covariants $V_G = V/\{ gv - v : v \in V, g \in G\}$ has codimension 1 in $V$ ( which is the case for my question ), then the ring of invariants $k[V]^{G}$ is a polynomial ring.

There's another paper: "Invariants of some Abelian $p$-groups in characteristic $p$" by Mara D. Neusel in Proc. AMS v. 125, no 7, pp. 1921-1931, showing that a similar result hold when codim$(V^G) = 2$ or codim$(V_G) = 2$, where $V^G = \{ v : v = gv \text{ for all } g \in G\}$.

Unfortunately neither applies in my case.