Here is a counterexample: A torus embedded in $\mathbb{R}^3$ has a fundamental class in $H_2$, but there are no points with neighborhoods homotopy equivalent to $S^2$.
It might be interesting to try to formulate a condition for the homology of pairs $(X,Y)$.
Edit: I think you may want to change your conjecture by demanding that there exist neighborhoods of a certain type that represent a set of homology cycles that span $H_k$. Otherwise, there doesn't seem to be a relationship between the neighborhoods and the elements of homology. It is also not clear why convexity makes an appearance - you may want to demand the neighborhood be contractible in $\mathbb{R}^m$, though.
With that in mind, here is a family of counterexamples that actually satisfy the conditions you specified: Embed the torus $(S^1)^n$ into some $\mathbb{R}^m$, for $n>2$ and sufficiently large $m$. Then for $k$ ranging from $2$ to $n-1$, the rational homology in degree $k$ has dimension $\binom{n}{k}$, but no cycles represented by spheres (or punctured Euclidean spaces).
In general, I think you should change "punctured Euclidean space" to "compact connected orientable manifold". Such objects look like holes if they aren't boundaries of higher-dimensional manifolds (and if you squint enough).
