Yes, there is 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 $D_0,\dots,D_{m-1}$, where each $E_i$ is equipped with a metric and 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
$$
L^2(E_i) = \ker(\Delta_i) \oplus \operatorname{im}(D_{i-1}) \oplus \operatorname{im}(D_i^*)
$$