Commenting on Peter Michor's answer, I want to say that we can however use the exponential map.
Is it true that the image of the exponential map is locally dense near the origin? If so, let us take open neighbourhoods $U\subset V$ of the identity such that $\overline{\exp(Vect_c(M))}\supset \overline{V}$. Given any $g\in V$, we can find $h_0\in U$ arbitrarily close to $id$ so that $gh_0^{-1}=f$ is the time $1$ map of a flow $\{f_t\}_{t\in[0,1]}$. Let $k=k(f)$ be the minimum integer such that $f_{1/k}$ belongs to $U$. Then $g$ is the product of $k+1$ diffeomorphisms in $U$: $$g=f_{1/k}\circ\cdots\circ f_{1/k}\circ h_0.$$
The problem here is that $\overline{U}$ is not compact so it doesn't seem clear that we can have a uniform bound on the choice of $k$ (though we still have some freedom in the choice of $h_0$).