The reason is that there is no mathematically rigorous construction of any *interacting* quantum field theory in four space-time dimensions to this date. Because of that, one has not been able so far to verify whether or not such a theory satisfies either the Garding-Wightman or the Haag-Kastler axioms. The only 4-dimensional QFT models known to satisfy these axiom schemes are non-interacting and even the academic $\lambda\phi^4$ QFT model has recently been shown by Aizenman and Duminil-Copin to be non-perturbatively trivial (that is, non-perturbative renormalization makes the model non-interacting).

Perturbative QFT models cannot satisfy either axiom scheme because the perturbative series for QFT remains formal in most models even after perturbative renormalization. More precisely, we know for physically relevant QFT models (e.g. QED) that the renormalized perturbative series is divergent. There is *partial* evidence that, for *some* (not all) such models, this series *may* be asymptotic, but that is all. All that leaving aside the fact that only the short-distance (i.e. "ultraviolet") renormalization procedure for perturbative QFT is mathematically well understood. The long-distance (i.e. "infrared") divergences in the perturbative S-matrix affect the very definition of scattering in QFT when there are massless particles and there is no complete, rigorous solution to this problem yet.

To sum up, QFT could be said "not to be fully axiomatized" simply because we still do not have enough mathematical control over relevant QFT models in order to check any of the proposed axiom schemes for QFT. My MO answer goes a bit deeper on the meaning and relevance of these schemes in such a scenario.