I assume that the question in the title is clear, so let me talk about its relevance:

According to theorem 4.3 in Heat Kernels and Dirac Operators the index theorem \begin{equation}\tag{1} \mathrm{ind}(D)=(4\pi)^{-n/2}\int_M\hat{A}(M)\wedge \mathrm{ch}(E/S) \end{equation} holds for every Dirac operator on a given Clifford module bundle (the set of Dirac operators is always non-empty as explained on page $117$). The theorem is proven for a Dirac bundle - a Clifford module bundle equipped with a Clifford connection and a compatible metric - and it follows that the theorem actually holds for all Dirac operators on $E$, because both the left-hand side and the right-hand side of $(1)$ are independent of the Dirac operator. But of course this leaves us with the question whether any Clifford module bundle can be extended to a Dirac bundle.