Let $R$ be a ring spectrum (in the world of EKMM $S$modules) and let $E$ be a smashing $R$module. Denote by $R_E$ the $E_*$localization of $R$. By a theorem of Wolbert (Theorem 2 in Classifying modules over $K$theory spectra), the derived category $\mathrm{Der}(R)[E^{1}]$ of $E_*^R$local $R$modules is equivalent to the derived category $\mathrm{Der}(R_E)$ of $R_E$modules. I wonder: is this equivalence induced by a Quillen equivalence $$ L_E \mathrm{Mod}(R)\simeq_Q\mathrm{Mod}(R_E)\; ? $$

This is an elaboration on Lennart's comment. This can be made to come from a Quillen equivalence. Here are the ingredients you'd usually need to show it. (Sorry, I don't have my copy of EKMM handy and so I can't provide theorem numbers.) The problem is that you haven't specified $R_E$ as an actual object yet rather than just as a homotopy type. There is a localization map $R \to R_E$. This can be chosen as a map of $S$algebras and with $R_E$ cofibrant as a right $R$module (which is possible up to equivalence, but not automatically satisfied). In this case, you can produce a Quillen equivalence. If you've got a representative for $R_E$ that's not a cofibrant $R$module, then you're typically going to have to replace it with an equivalent algebra $R'_E$ which is, and this will give a zigzag of Quillen equivalences $L_E Mod(R) \sim Mod(R'_E) \sim Mod(R_E)$. (Really, all we'll need is a "flatness" property.) Now let's give some details of the proof. Unless I'm mistaken, the ordinary model structures on $Mod(R)$ and $Mod(R_E)$ are lifted from $Mod(S)$: a map is a weak equivalence or a fibration if and only if it is so in $S$modules. For this reason, the forgetful map is automatically a right Quillen functor. Then $R_E \wedge_R ()$ is a left Quillen functor, preserving cofibrations and weak equivalences. Now let's talk about the localization. This is usually taken to be a left Bousfield localization on the model category level: $L_E Mod(R)$ has the same underlying category, and the same underlying cofibrations, as $Mod(R)$, but new weak equivalences (the $E$equivalences) and new fibrations. Since it's the same underlying category, we still have the forgetful functor from $Mod(R_E)$ to $L_E Mod(R)$, with left adjoint $R_E \wedge_R ()$. The left adjoint still preserves cofibrations. More, it preserves weak equivalences. Since this is a smashing localization, a map $X \to Y$ is an $E$equivalence if and only if it's an equivalence after taking derived smash product over $R$ with $R_E$. Since $R_E$ is cofibrant, smash products represent derived smash products $\wedge^{\mathbb L}_R$. Therefore, $X \to Y$ is an $E$equivalence if and only if $R_E \wedge_R X \to R_E \wedge_R Y$ is a weak equivalence. Now we finally need to show that this is a Quillen equivalence. Suppose $X$ is a cofibrant $R$module and $Y$ is a fibrant $R_E$module. We already showed that a map $X \to Y$ of $R$modules is an $E$equivalence if and only if the map $R_E \wedge_R X \to R_E \wedge_R Y$ is an equivalence. The unit map $R_E \wedge_R Y \to Y$ is always an equivalence because $Y$ is $E$local and this is a smashing localization. Therefore, $X \to Y$ is an equivalence in $L_E Mod(R)$ if and only if its adjoint $R_E \wedge_R X \to R_E \wedge_R Y \to Y$ is an equivalence in $Mod(R_E)$, making this into a Quillen equivalence. 

