Semidefinite programs (SDPs) have many applications in the field of controls. To improve scalability, it is important to exploit the inherent sparsity when solving SDPs. In this paper, we develop a new spectral bundle algorithm that solves sparse SDPs without introducing additional variables. We first apply chordal decomposition to replace a large positive semidefinite (PSD) constraint with a set of smaller coupled constraints. Then, we move the coupled constraints into the cost function via exact penalty. This leads to an equivalent non-smooth penalized problem, which can be solved by bundle methods. We present a new efficient spectral bundle algorithm, where subgradient information is incorporated to update a lower approximation at each iteration. We further establish sublinear convergences in terms of objective value, primal feasibility, dual feasibility, and duality gap. Under Slater's condition, the algorithm converges with the rate of $\mathcal{O}(1/\epsilon^{3})$, and the rate improves to $\mathcal{O}(1/\epsilon)$ when strict complementarity holds. Our numerical experiments support the theoretical analysis.