I don't have a complete picture on what has been done on gauge theories in the context of axiomatic QFT. Some time ago, there was some discussion in the PhysicsForums thread
How does a gauge field lead to charge superselection?
which also gives specific references. Some of what I said there is a
bit inaccurate, due to my then still limited understanding.
The Wightman axioms are known not to cover any of the fundamental quantum fields theories (QED, QCD, and the standard models), as these are gauge theories and the Wightman axioms are not
applicable to these. All realistic 4D field theories are either nonrenormalizable (so that not even perturbation theory defined them well), or gauge theories (for which the Wightman axioms are
inappropriate as they don't allow gauge-dependent charged states). There have been attempts to repair this (notably by Strocchi); some of this can be found in the recent book
An introduction to non-perturbative foundations of quantum field theory,
Oxford Univ. Press, 2013.
Description of noncommutative theories and matrix models by Wightman functions,
J. Math. Phys. 45, 4980 (2004)
(preprint version, significantly different:
But the current understanding is partial only. Thus we currently have no adequate system of axioms for QFT, only a number of ideas how it could possibly look like.
The main reason why modifications are needed is that gauge fields are unobservable, but the standard Wightman axioms cover only the (expectation values of products of generators of the) C^*-algebra of bounded observable fields, which is a small subalgebra of the algebra generated by the gauge fields. It does not even contain charged operators.
This small algebra does not contain enough local observables to give a Wightman theory. The reason is that gauge invariant fields made from gauge fields are necessarily nonlocal, while the Wightman axioms assume implicitly that the algebra is generated by local fields.
The gauge fields themselves are local but on the bigger algebra generated by them it is impossible to define a positive definite inner product, and hence a C^* algebra. The Hilbert space is replaced by an indefinite inner product space, whence the bounded operators of the bigger algebra do not form a C^*-algebra structure.
This makes it difficult to figure out how to model a gauge theory within the Wightman framework. Partial work involves the classification of superselection sectors, which defines charged representations, but only leads to global gauge transformations, not to local gauge theories.
Note that among the algebraic tools, closest to the standard model are in fact not the nonperturbative Wightman axioms but the perturbative Epstein-Glaser approach to quantum field theory; see http://en.wikipedia.org/wiki/Causal_perturbation_theory and the book
Quantum Gauge Theories: A true ghost story
But this is mathematically incomplete as it is just renormalized perturbation theory, without well-defined observables.