How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$? Suppose $A$ and $B$ are $E_{\infty}$ rings, then $\mathrm{Mod}(A)$ and $\mathrm{Mod}(B)$ are $E_{\infty}$ monoidal categories (left modules over those rings). We can ask about $E_n$ colimit-preserving morphisms between these two categories. How do we characterize them? If $n=\infty$ we just have maps from $A$ to $B$ and if we don't ask for any monoidal compatibility I guess we just get bimodules over $A$ and $B$, but what are the intermediates? Maybe we can also ask for a characterization of $E_n$ maps of the category of $E_n$ modules over $E_n$ rings?
Edit: Thanks to Dylan's answer below the first question is solved. However I believe the second question is still not answered, which is about studying $E_n$ maps between categories of $E_n$ modules over $E_n$ rings.
 A: By Corollary HA.4.8.5.20, the functor from $\mathbb{E}_{n+1}$-algebras to $\mathbb{E}_n$-monoidal categories and colimit-preserving, $\mathbb{E}_n$-monoidal functors is fully faithful. (Notice that the condition of being "linear over Sp" is automatic from exactness). So, indeed, you'd get just $\mathbb{E}_{n+1}$-algebra maps $A \to B$.
Added:
I just wrote down the reference, but of course, as Ben-Zvi points out, this should be relatively intuitive. First of all, building a functor $F:\mathsf{RMod}_A \to \mathcal{C}$ means that I'd better tell you how $A=\mathrm{End}_A(A)$ acts on $F(A)$, and you can kinda see that that is all I need to do if $F$ commutes with colimits (because I can start resolving by free modules and I understand what to do with all the maps). In other words, I need to provide an algebra map $A \to \mathrm{End}_{\mathcal{C}}(F(A))$. So now we believe that $A \mapsto \mathsf{RMod}_A$, viewed as a functor to categories equipped with a distinguised object, has a right adjoint given by $(\mathcal{C},c)\mapsto \mathrm{End}(c)$.
The functor $A \mapsto \mathsf{RMod}_A$ turns out to be symmetric monoidal when one views its values as landing in, say, stable, presentable categories, and so its right adjoint will be lax symmetric monoidal. It follows that we get an induced adjunction on categories of algebras, and that the endomorphism object of the unit in an $\mathbb{E}_n$-monoidal category is canonically $\mathbb{E}_{n+1}$-monoidal (you can think of that like: you get an $\mathbb{E}_1$-algebra structure from composing, and then an $\mathbb{E}_n$-algebra structure from tensoring, so together you get an $\mathbb{E}_{n+1}$-algebra structure).
All together, then, to specify an $\mathbb{E}_n$-monoidal functor $\mathsf{RMod}_A \to \mathcal{C}$, I just need an $\mathbb{E}_{n+1}$-algebra map $A \to \mathrm{End}_{\mathcal{C}}(1)$.
There's some other discussion about how to describe maps between categories of $\mathbb{E}_n$-modules, but I think that's not as clear to me... Yes, it's true that the endomorphisms of the unit in the category of $\mathbb{E}_n$-modules is the $\mathbb{E}_n$-center, but if the source of our functor is some other category of $\mathbb{E}_n$-modules (as opposed to ordinary modules over something $\mathbb{E}_{n+1}$), then our argument above doesn't apply.
A: To clarify my comment: suppose you're interested in lax $E_n$-monoidal functor (so that indeed, you get coherent maps $F(M)\otimes F(N)\to F(M\otimes N)$ which aren't necessarily equivalences, similarly for $1\to F(1)$) that preserve colimits, then the category of such functors is essentially $Alg_{E_n}(Fun^{ex}(Mod_A^\omega,Mod_B))$ where $Fun^{ex}(Mod_A^\omega, Mod_B)$ has a symmetric monoidal structure induced by the Day convolution structure on $Fun(Mod_A^\omega, Mod_B)$.
Then you get a symmetric monoidal identification $Mod_{A^{op}\otimes B} \simeq Fun(Mod_A^\omega,Mod_B)$ which, since $A$ is $E_\infty$, can be thought of as $Mod_{A\otimes B}$.
Therefore lax $E_n$-monoidal colimit preserving functors $Mod_A\to Mod_B$ can be thought of as $E_n-A\otimes B$-algebras.
Of course, these rarely classify strict monoidal functors (those where $F(M)\otimes F(N)\to F(M\otimes N)$ is an equivalence, and $1\to F(1)$ is as well). In fact they do if and only if this $A\otimes B$-module is $B$ with structure induced from an $E_{n+1}$-map $A\to B$: Tyler's answer shows this.
I made a silly comment below Tyler's answer (then deleted, as I realized my mistake), which is perhaps noteworthy: if you try to take this approach and check when, for an $A\otimes B$-module $R$, the associated lax $E_n$-monoidal functor is actually strict, you'll get two conditions: one on the map $F(M)\otimes F(N)\to F(M\otimes N)$ and one on the unit. The second one is easy to forget (which is what happened in my silly comment), and if you do, you'll get something like an $E_n-B$-algebra such that the structure map $B\to R$ is an epimorphism.
This is why for instance $Mod_\mathbb Q\to Sp$ is almost symmetric monoidal, although it is only lax: you get the condition on tensor products but not the one on the unit.
