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 $h_0$).