There are, at least, many examples where this is false. For instance, Hitchin showed (I think this was his thesis) that there is a sequence of metrics $\{g_k\}_{k=1}^{\infty}$ on the three sphere such that the kernel of the Dirac operator associated to $g_{k}$ has dimension at least $k.$ It is stated as a conjecture in these notes http://www.mathematik.uni-regensburg.de/ammann/talks/11BerlinSFB.pdf that this is a generic phenomena for spin manifolds of dimension at least three.

In the case where $E=F=M\times \mathbb{R},$ I don't have any examples in mind off the top of my head, but I could imagine one could construct a family of metrics $\{g_{k}\}$ and a family of functions $f_{k}:M\rightarrow \mathbb{R}$ such that
the sequence of elliptic operators
\begin{align}
D_{k}=\Delta_{g_{k}} + f_{k}
\end{align}
has arbitrarily large kernel.

With this in mind, it's hard to imagine a circumstance where what you are asking is true. A better question might be: for some sensible elliptic operator that depends on a parameter, like the Dirac operator on a spin manifold depending on a Riemannian metric, can you find sensible geometric bounds on the Riemannian metric under which this kernel has a uniform bound.