For any multiplicative cohomology theory $E^*$, it seems to be that everything in the long exact sequence of a pair and in the Mayer–Vietoris sequence is a module map over $E^*$ of the big space. That is, if $(X,A)$ is a CW-pair or $(X;U,V)$ an excisive CW-triad, in the long exact sequences $$\cdots \to E^*(X,A) \longrightarrow E^* X \longrightarrow E^*A \longrightarrow E^{*+1}(X,A) \to \cdots$$ and $$\cdots \to E^* X \longrightarrow E^* U \times E^* V \longrightarrow E^*(U \cap V) \longrightarrow E^{*+1} X \to \cdots,$$ each object is a module over the ring $E^* X$ in such a way that all maps are $E^*X$-module homomorphisms. This is easily checked on the cochain level for singular cohomology, but seems to follow generally from the axioms. Assuming this is right, it must be well known, but it isn't in the first half-dozen books I've checked. What is a reference for this?
Details: In Hatcher's K-theory book, he shows that in the reduced K-theoretic long exact sequence of a compact CW pair $(X,A)$, all maps are actually $\tilde K^* X$-module homomorphisms; the non-obvious thing is that $\tilde K^*(X/A)$ admits a $\tilde K^*X$-module structure preserved by the connecting map from $\tilde K^{*-1} A$.
This then trivially extends to a result on unreduced K-theory. But Hatcher's proof seems to apply equally well to any multiplicative reduced cohomology theory: one uses naturality of the cross product $$\widetilde E^* X \otimes \widetilde E^* (-) \longrightarrow \widetilde E^*(X \wedge -)$$ and commutativity of a diagram $$ \require{AMScd} \begin{CD} A @>>> X @>>> X \cup C_* A @>>> \Sigma A\\ @VVV @VVV @VVV @VVV \\ X \wedge A @>>> X \wedge X @>>> X \wedge (X \cup C_* A) @>>> X \wedge \Sigma A \end{CD} $$ of spaces, where all maps are induced from the diagonal map $X \to X \wedge X$. The naturality of the outer product is axiomatic and the commutativity of the diagram is purely a property of spaces, so there's not much that could go wrong.
For the Mayer–Vietoris sequence it's clear all the graded groups in the sequence are modules over $\widetilde E^* X$ by restricting along the inclusions, and the only map which could potentially not preserve this structure is the connecting map. But this is defined as the composition $$\widetilde E^*(U \cap V) \overset\delta\longrightarrow \widetilde E^{*+1}\Big(\frac U{U\cap V}\Big) \overset \sim \longleftarrow \widetilde E^{*+1}(X/V) \longrightarrow \widetilde E^{*+1} X,$$ where the first map is the coboundary for the pair $(U,U\cap V)$, which, like the third map, we've already shown enjoys this structure, and the second is induced by the obvious homeomorphism.