set theory - Set theories that do require the existence of urelements?

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 set-theories-with-urelements 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 Section 3, we construct a model of set theory with no nontrivial injective abelian groups. It is a permutation model in which the atoms (= urelements) form a proper class;

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.

Definitions and Conventions. The theory ZFA is a modification of ZF allowing atoms, also
known as urelements. See Jech [4] for a precise definition. A model of ZFA may have a proper
class of atoms; however, for this paper we redefine ZFA to include an axiom which says that
the class of atoms is a set (always denoted by A).