Here is the main theorem of "A Vietoris Mapping Theorem for Homotopy" by S. Smale:

THEOREM: Let $f:X\to Y$ be proper and onto, where $Y$ and $X$ are

- $0$-connected
- separable
- locally compact
- metric.

Assume $X$ is $LC^n$ (which I presume means locally $n$-connected) and
that for each $y\in Y$, $f^{-1}(Y)$ is $LC^{n-1}$ and
$(n-1)$-connected. Then

- $Y$ is $LC^n$ and
- $f$ is an $n$-equivalence.

Some comments:

- $f$ is proper means the preimages of compact sets are compact; this is automatic if $X$ is compact and $Y$ is Hausdorff.
- the local conditions on $X$ and $Y$ are satisfied if $X$ and $Y$ are CW complexes.

In your situation, if your $f$ satisfies the conditions of this theorem for all $n$, then we deduce that $f$ is an $n$-equivalence for all $n$ and hence that $E$ is weakly contractible because $C$ is contractible.