I am looking for an axiomatic set theory that not only admits the existence of urelements/atoms (via twosortedness or an additional unary predicate) but requires it, e.g. by an axiom like "for each set there is an equipollent set of urelements" (= "there are arbitrarily many urelements"). Any references?
Nominal logic is based on the FrænkelMostowski permutation model of set theory. In particular Nominal logic has a freshness axiom that states ${\forall}x. {\exists}a \in \mathbb{A}. a \# x$, where $\mathbb{A}$ is the set of atoms and For reference, see the work by Andrew Pitts and Murdoch Gabbay. For example "A New Approach to Abstract Syntax Involving Binders". 


Your question is equivalent to asking whether the urelements, or atoms, can form a proper class. This axiom is consistent with ZFA, but usually ZFA is introduced so as to not insist on this (and indeed, not insist on any atoms at all). I believe that many (or most) of the other standard settheorieswithurelements also allow this. Andreas Blass has an article here, where he investigates the connection between some theorems in homological algebra and the Axiom of Choice. In his introduction, he states:
In contrast, sometimes it is useful to have only a set of atoms, as witnessed by Eric Hall's article, which contains the following remark.



In the final end I found such a theory, it's called ZFCUA (= Zermelo Frankel set theory with the axiom of choice and unlimited atoms), see Faithful Representation in Set Theory with Atoms by Harvey Friedman. The relevant axiom is #11: "There is no set consisting of all atoms." A consequence of this axiom is, that there definitely are atoms (since the empty set is a set) and furthermore, that there are so many, that the collection of all of them is not a set but a proper class. 


I don't know if this is exactly what you're looking for, but it's a theorem of NFU that $\mathcal{P}(V) < V$which has as a corollary not only that there are atoms, but that the set of atoms is equipollent with the universe. (A somewhat more disquieting way of putting this is that there are more atoms than there are sets.) 

