Direct axiomatization of ordinal and cardinal numbers
Are these papers of Takeuti the sort of thing you want?
MR0086751 (19,237e) 02.0X Takeuti, Gaisi, On the theory of ordinal numbers. J. Math. Soc. Japan 9 (1957), 93–113.
MR0099918 (20 #6354) 02.00 Takeuti, Gaisi, On the theory of ordinal numbers. II. J. Math. Soc. Japan 10 1958 106–120
MR0197302 (33 #5467) 02.18 Takeuti, Gaisi, A formalization of the theory of ordinal numbers. J. Symbolic Logic 30 1965 295–317
I would like to suggest the theory of "ordinal algebras" and "cardinal algebras". There are books with the same title by Tarski.
Mathscinet review of the book cardinal algebras:
This book is an axiomatic investigation of the novel types of algebraic systems which arise from three sources: the arithmetic of cardinal numbers; the formal properties of the direct product decompositions of algebraic systems; the algebraic aspects of invariant measures, regarded as functions on a field of sets.
Mathscinet review of the book ordinal algebras:
An ordinal algebra is an additively written, but not ordinarily commutative, associative system, equipped with a suitably axiomatized operation of simply infinite addition, $∑^∞_1a_ν$, and an operation of conversion, $a^∗$. (Infinite addition is characterized and employed "only insofar as [it is] involved in the study of finite addition''.) The additive theory of order types provides a familiar application, although far from exhausting the interest of the theory. The theory of ordinal algebras differs from that presented in the author's "Cardinal algebras'' principally in the non-commutativity of addition.
This is not exactly what you asked for, but there is an interesting axiomatization of Ordinals + Sets of Ordinals, which turns out to be precisely equiconsistent with ZFC.
- Peter Koepke, Martin Koerwien, The Theory of Sets of Ordinals.
Basically, if one is committed to the ordinals and having certain kinds of sets of ordinals, then you can build Gödel's constructible universe $L$ and simulate a model of ZFC that way.