Yes, there is a Hodge decomposition for elliptic complexes on compact oriented Riemannian manifolds.
Let $(M,g)$ be a compact oriented Riemannian manifold and let $$ 0 \to \Gamma(E_0) \to \Gamma(E_1) \to \dots \to \Gamma(E_m) \to 0 $$ be an elliptic differential complex with the arrows given by the differential operators $D_0,\dotsc,D_{m-1}$, where each $E_i$ is equipped with a metric and a compatible connection. Thus we have formal adjoints and can form the Laplacians: $$ \Delta_i := D_i^* \circ D_i + D_{i-1} \circ D_{i-1}^*: \Gamma(E_i) \to \Gamma(E_i). $$ Then we get the Hodge decomposition $$ L^2(E_i) = \ker(\Delta_i) \oplus \operatorname{ran}(D_{i-1}) \oplus \operatorname{ran}(D_i^*). $$
I learnt this from our own Liviu Nicolaescu's Lectures on the Geometry of Manifolds. More precisely, this is Chapter 10.4.3 in the notes (my version is from 9. Sep 2018).
