If I read your updated question correctly, you are asking whether people have considered non-linear modifications of quantum mechanics in order to accommodate interacting QFTs. I'm sure someone, somewhere has, but that's certainly not mainstream thought in QFT research, either on the mathematics or theoretical physics sides. Consider the analogous question in the quantum mechanics of particles: do non-linear equations of motion require a non-linear modification of quantum mechanics? The answer is most certainly No.
Without going into generalities, the Hydrogen atom and the double-well potential are prominent examples of systems with non-linear (Heisenberg) equations of motion that live perfectly well within the standard quantum formalism (states form a linear Hilbert space, observables are linear operators on states, time evolution is unitary on states in the Schroedinger picture and conjugation by unitary operators in the Heisenberg picture). When going from particle mechanics to field theory, what changes is the number of space-time dimensions, not the type of non-linearities in the equations of motion. So there is no mathematical reason to expect a non-linear modification of quantum mechanics in the transition.
Now, a few words about your intuition regarding states as solutions to the equations of motion. Unfortunately, it is somewhat off the mark. As you should be aware, relativistic QFT is usually discussed in the Heisenberg picture. This means that it is the field operators $\hat{\phi}(t,x)$ that obey the possibly non-linear equations of motion. For example, $\square\hat{\phi}(t,x) - \lambda{:}\hat{\phi}^3(t,x){:}=0$, where $\square$ is the wave operator and the colons denote normal ordering. On the other hand, states are just elements $|\Psi\rangle$ of an abstract Hilbert space (with the vacuum state $|0\rangle$ singled out by Poincaré invariance), entirely independent of spacetime coordinates. At this point, it should be clear why states have nothing to do with the equations of motion.
Your intuition is not entirely without basis, though. Spelling it out, also shows how the standard formalism of QFT (Wightman or any related one) already accommodates non-linear interactions. One can define the following hierarchy of $n$-point functions (sometimes called Wightman functions):
\begin{align}
W^0_\Psi &= \langle 0|\Psi\rangle \\
W^1_\Psi(t_1,x_1) &= \langle 0|\hat{\phi}(t_1,x_1)|\Psi\rangle \\
W^2_\Psi(t_1,x_1;t_2,x_2) &= \langle 0|\hat{\phi}(t_1,x_1)\hat{\phi}(t_2,x_2)|\Psi\rangle \\
& \cdots
\end{align}
It is a fundamental result in QFT (known under different names, such as the Wightman reconstruction theorem, multiparticle representation of states, or simply second quantization) that knowledge of all the $W^n_\Psi$ is completely equivalent to the knowledge of $|\Psi\rangle$.
These Wightman functions, by virtue of the Heisenberg equations of motion, satisfy the following infinite dimensional hierarchical system of equations
\begin{align}
\square_{t,x} W^1_\Psi(t,x) &= \lambda W^3_\Psi(t,x;t,x;t,x) + \text{(n-ord)} \\
\square_{t,x} W^2_\Psi(t,x;t_1,x_1) &= \lambda W^4_\Psi(t,x;t,x;t,x;t_1,x_1) + \text{(n-ord)} \\
\square_{t,x} W^2_\Psi(t_1,x_1;t,x) &= \lambda W^4_\Psi(t_1,x_1;t,x;t,x;t,x) + \text{(n-ord)} \\
& \cdots
\end{align}
I'm being a bit sloppy with coincidence limits here. The Wightman functions are singular if any two spacetime points in their arguments coincide, the terms labeled (n-ord) represent the necessary regulating subtractions to make this limit finite. This necessary regularization also explains why the non-linear terms in the equations of motion needed normal ordering.
If $\lambda=0$, the theory is non-interacting, then each of the above equations for the $W^n_\Psi$ becomes self-contained (independent of $n$-point functions of different order) and identical to the now linear equations of motion. At this point it should be clear how your intuition does in fact apply to the states of a non-interacting QFT. States $|\Psi\rangle$ can be put into correspondence with multiparticle "wave functions" solving the linear equations of motion (which are actually the Wightman functions $W^n_\Psi$).
Finally, when it comes to trying to construct models of QFT, people usually just concentrate on the Wightman functions associated to the vacuum state, $W^n_0 = \langle 0|\cdots|0\rangle$, which are sufficient to reconstruct the corresponding $n$-point functions for all other states. In short, the standard approaches to constructive QFT already incorporate non-linear interactions in a natural way. And non-linear modifications to the quantum mechanical formalism are simply a whole different, independent topic.
Best Answer
As Abdelmalek Abdesselam pointed in his comment to the OP, the axiomatic approach to QFT is rather concerned with answering the question "what is a quantum field?". This is stated right at the Preface of the book of Streater and Wightman, PCT, Spin and Statistics, and All That. More precisely, it lists a minimal set of desiderata for a reasonable concept of a "quantum field", and deals with the consequences of such requirements, which can be thought of as "structural" or "model-independent". It leaves open the question of which fields used in physics actually satisfy these requirements, apart from checking that free fields do. This is important as a first relevance check, since free fields are the easiest to mathematically control - if not even a free field complies with a list of axioms for QFT, then these axioms are useless as they stand and should be modified. More generally, any list of mathematical axioms applied to a class of physical theories is bound not to be set in stone, since our knowledge of Nature is always approximate due to our limited experimental precision. One should rather see such axioms (and their limitations in the given context) as general physical principles whose robustness has to be constantly tested, in addition to forming a well-defined mathematical framework by themselves. Such is the way of natural sciences.
In that respect, it must be remarked that indeed not all physically relevant fields satisfy the Garding-Wightman axioms - most notably, fields acting in Krein spaces (i.e. "Hilbert" spaces with a possibly indefinite scalar product), such as the electromagnetic potential in a covariant gauge, do not. The corresponding axiom for vacuum expectation values that fails is that of positivity. There are ways to extend the Wightman formalism to such fields, but the results are nowhere near as mighty of even as rigorous, since positive definiteness of the scalar product is a powerful constraint. Another tricky example are perturbative (renormalized) quantum fields, since these are formal power series in the coupling constant (convergence of the renormalized perturbative series is usually expected to fail). One must in this case keep track of the order-by-order structure of all series involved, and to define certain concepts invoked by the Garding-Wightman axioms such as positivity is far from trivial. Regarding renormalization, the main conceptual challenge is not so much the ultraviolet problem (which is rigorously well understood on a formal perturbative level), but the infrared problem, which plagues all interacting QFT models with massless fields and is not completely understood on a rigorous level, even in formal perturbation theory. And all that before even considering how to extract some non-perturbative definition of the model from perturbative data by employing e.g. some generalized summability concept for the perturbative series.
This order of doing things - namely, stating a list of general desiderata and then checking if models comply with it or not - is what is often called a "top-down" approach to QFT. What people such as Folland try to do is rather a "bottom-up" approach: trying to make sense of the formal procedures physicists actually use in (perturbative) QFT. A more successful way of doing what Folland tries to do is achieved by the so-called perturbative algebraic QFT, which draws ideas from the Haag-Kastler algebraic approach to QFT (which is a different axiomatic scheme from that of Garding and Wightman, aiming at aspects which do not depend on the particular Hilbert space the fields live in) in order to mathematically understand formal renormalized perturbative QFT (summability of the perturbative series has not been addressed yet in this approach). The so-called constructive approach to QFT also fits in such a "bottom-up" philosophy: it tries to build QFT models by means other than perturbation theory, in view of its expected divergence in relevant cases (Abdelmalek can tell you far more than me on that). Once one has obtained the model, one may try to e.g. check the Wightman axioms (or some suitable modification thereof) in order to see if the model obtained complies with that particular notion of QFT, which is undoubtedly relevant.
To sum up, I would say that both approaches are important and complement each other.