The question here is how to organize a response. Names can be given, but I hope I can give some conceptual order. Hopefully other answers will find other ways to organize their response.
The Wightman axioms are classic. I take the approach here of organizing how other approaches sit with the Wightman axioms, even though they may not be axiomatic. A useful critique of the Wightman axioms can be found in Streater R F, Rep. Prog. Phys. 1975 38 771-846. More recent is the assessment of Fredenhagen, Rehren, and Seiler, “Quantum Field Theory: Where We Are” in http://arxiv.org/abs/hep-th/0603155, which I recommend. In my scheme here, however, the approach you particularly ask about, formalizations of the renormalization group, do not figure, because, as you say, they have a completely different starting point. I’d say that the starting point is perhaps the concept of Feynman integrals rather than the renormalization group in itself, but I’d also say that that’s a quibble.
There is a large question of what we hope to achieve by axiomatizing. (1) We can loosen the axioms, so that we have more models, some of which might be useful in Physics, but we have to figure out which on a case-by-case basis. This makes Engineering somewhat quixotic. (2) We can tighten the axioms, with the ambition that all the models are useful in Physics, but some Physically useful models might be ruled out. Mathematicians are often happy to work with axioms that a Physicist would consider too tight.
So, the Wightman axioms, more-or-less in Haag’s presentation in “Local Quantum Physics”:
The state space
(a) is a separable Hilbert space. There are people trying to use non-associative algebras, amongst other things.
(b) which supports a representation of the Poincaré group. There are people doing QG, QFT on CST, and many ways of breaking Lorentz symmetry at small scales.
(c) There is a unique Poincaré invariant state. Thermal sectors don’t satisfy this. Non-unique vacuums are an old favorite, but the vacuum state is pervasive in Particle Physics.
(d) The spectrum of the generator of translations is confined to the closed forward light-cone. This is an elephant, IMO. The underlying reason for this is “stability”, which has no axiomatic formulation. The belief that the spectrum condition is necessary for stability may rely on classical thinking, particularly on the primacy of the Hamiltonian or Lagrangian. Feynman integrals for loops introduce negative frequencies, however, so there’s something of a case against it.
(which, implicitly, correspond in some way to statistics of experimental data)
(a) Are operator-valued distributions.
People have introduced other Generalized function spaces. Haag-Kastler tightens this, to bounded operators, but the mapping from space-time regions to operators is looser. In Particle Physics, the S-matrix, which discusses transitions between free field states on time-like hyperplanes at t=+/-infinity, has been the supreme observable for decades: trying to reconcile this with the Lorentz invariant operator-valued distributions of the Wightman axioms pretty much killed the latter. Condensed matter Physics, optics, etc., take correlation functions quite seriously, which seems to me to be at the heart of the split between Particle and other Physicists. Another elephant.
(b) Are Hermitian.
There’s a complex structure. People have also introduced quaternions in various ways.
(c) The fields transform under the Poincaré group. This goes with 1b.
(d) The observables are jointly measurable at space-like separation, but in general are not jointly measurable at time-like separation. Stepping away from the Poincaré group almost always results in violation of this axiom. Random fields, which are always jointly measurable operator-valued distributions, and the differences between them and QFT, are something I have published on.
The Haag-Kastler approach to some extent brings the states and observables into the single structure of von Neumann algebras, but essentially the distinction of linear operators and their duals remains. Refusing to split the world into states and observables, which we might call “holism”, makes Physics almost impossible. There’s always the question of exactly where the Heisenberg cut should be put, but pragmatically we just put it somewhere. Bell tries to square that circle while still doing Physics in his ‘Against “Measurement” ’, and Bohm as much as left Physics behind. There are people trying to do that kind of thing, but I find very little of it useful.
Returning back to earth, there’s also a question of how we deform a system that we have managed to construct so that it’s significantly different in interesting ways. This isn’t in the axioms, but the standard has been to deform the Hamiltonian or Lagrangian. Both methods, however, require a choice of one or two space-like hypersurfaces, which goes against the spirit of the Poincaré group. Algebraic deformations, the other known alternative (others?), have hardly left the ground because the constraints of positive energy, microcausality, and the primacy of the S-matrix have hitherto ruled them out (I have also published on this, based on Lie fields from the 1960s). If we deform the algebra of observables instead of the dynamics, the question arises of what "stability" might be.
There is of course a question whether one ought to start from the Wightman axioms at all, but one has to choose somewhere. Then, with Lee Smolin, one has to set off into the valleys, hoping to find a bigger hill.