Yes, there is a Hodge decomposition for elliptic complexes on compact *oriented* Riemannian manifolds.

**$L^2$-version**. 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](https://doi.org/10.1142/9789812770295)*. More precisely, this is Chapter 10.4.3 in the notes (my version is from 9. Sep 2018).

**EDIT:** As per OP's comment, I am adding the smooth version I know of.

**$C^{\infty}$-version**. Consider the total vector bundle $E := \bigoplus_{i=0}^{m} E_i$ and the Laplacian $\Delta = D \circ D^* + D^* \circ D$, i.e. $\Delta$ is just a tuple of Laplacians $\Delta_i: \Gamma(E_i) \to \Gamma(E_i)$. The following statements are equivalent:

 (1) $\Delta$ is Fredholm and $\Gamma(E) = \ker(\Delta) + \operatorname{ran}(\Delta)$ (note, this is just a sum, not a direct sum);

 (2) The complex itself is Fredholm and satisfies the Hodge decomposition:
$$
\Gamma(E) = \ker(\Delta) \oplus \operatorname{ran}(D) \oplus \operatorname{ran}(D^*)
$$
This can be found in Chapter 1 of [van den Ban and Crainic's lectures on *Analysis on Manifolds*](https://webspace.science.uu.nl/~ban00101/geoman2017/AS-2017rev.pdf), more precisely exercise 1.3.15, if you don't mind following the exercises and working it out from the preceding theory, but I don't know a more self-contained source.