My guess is this is the kind of algebra you don't care about (since they aren't subrings of real-valued functions), but algebras of the form $\mathbb{R}[x]/x^n$ will have your property. When looking for a flow corresponding to a derivation $\xi$, consider the formal power series
$$ \sum_i \frac{t^i\xi^i}{i!} $$
The problem, of course, is showing that this power series converges in the algebra of appropriate endomorphisms. However, a lazy trick for guaranteeing its convergence is to hope that $\xi^n=0$ for some $n$. This happens for every derivation in the ring $\mathbb{R}[x]/x^n$, or more generally when the nilradical is maximal.
Edit: As is pointed out in the comments, this is a correct claim but the wrong reason. The real reason flows exponentiate in these rings, as well as all finite dimensional ones, is that a derivation is given by a matrix (after some choice of basis), and all matrices can be exponentiated (that is, the above series converges).