Let $X$ be the wedge of infinitely many circles (equipped with the CW topology). Every vector bundle $\xi$ over $X$ is a summand of a trivial bundle, namely it is $\xi\oplus\xi$ is trivial because any vector bundle over a circle has this property (alternatively, one could appeal to the fact that $X$ is homotopy equivalent to a smooth manifold, as explained in my comment above).
Suppose there is a compact space $K$ and a homotopy equivalence $f: K\to X$. The image $f(K)$ is compact, so $f(K)$ lies in a finite subcomplex $X_0$ of $X$, i.e. $X_0$ is a wedge of finitely many circles. Thus any loop on $X$ is freely homotopic to a loop in $X_0$, which is false (because a circle that forms the wedge can be homotoped into $X_0$ only if it lies in $X_0$.