This is going to basically be a cross-post of a MSE question: http://math.stackexchange.com/questions/147146/mayer-vietoris-implies-excision. I suspect that the answer to this question will turn out to be fairly straightforward, but (a) I can't figure it out myself, (b) it's not in any of the obvious references, (c) it hasn't gotten any answers on MSE, and (d) the question honestly is relevant to my research.

Let $H_n$ be a sequence of functors from the category of locally compact Hausdorff spaces to the category of abelian groups which satisfy all of the usual Eilenberg-Steenrod axioms for a generalized homology theory, except the excision axiom is replaced by a Mayer-Vietoris axiom: whenever $X$ is the union of two closed subspaces $A$ and $B$ there is a (functorial) exact sequence

$$\to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \to H_{n-1}(A \cap B) \to$$

According to the slogan "Mayer-Vietoris is equivalent to excision", one ought to be able to prove an excision theorem for $H_n$. This probably takes the form of a long exact sequence

$$\to H_n(A) \to H_n(X) \to H_n(X - A) \to H_{n-1}(A) \to$$

where $A$ is a closed subset of $X$, though if this isn't the right formulation I welcome corrections. I know how to go the other direction (excision implies Mayer-Vietoris), but this direction eludes me. Can anyone help?