I would think the answer is yes and follows by applying "[1.0.1] Theorem" on page 2 [here][1] for the finite Borel measures $\Re^+f(g)dg$, $\Re^-f(g)dg$, $\Im^+f(g)dg$, $\Im^-f(g)dg$ on $G$, where $\pm$ stands for positive and negative part. Note that Frechet spaces satisfy the conditions there.

**EDIT.** As the OP pointed out, the result is stated for compactly supported functions. I think an extension to integrable functions is straightforward when $V$ is a complete metric space like a Frechet space. Indeed, one can choose a sequence $f_n\in C_c^\infty(G)$  converging to $f$ in $L^1(G)$. The last line in the quoted theorem implies that the vectors $w_n:=\pi(f_n)v\in V$ form a Cauchy sequence by $\mathrm{dist}(w_n,w_m)\leq 2|f_n-f_m|_{L^1(G)}$. Now by completeness $(w_n)$ converges to some $w\in V$ and this $w$ would be $\pi(f)v$ by definition.

  [1]: http://www.math.umn.edu/~garrett/m/fun/Notes/07_vv_integrals.pdf