Is there any theorem that suggests that QM+SR has to be an operator theory?

What you call an operator theory is usually called the Heisenberg picture of quantum mechanics. What you call a wave function theory is usually called the Schroedinger picture of quantum mechanics.

It is well-known that for every quantum mechanical model, the Heisenberg picture and the Schroedinger picture are fully equivalent through a dual description, as long as on doesn't consider time correlations.

This therefore also holds for quantum field theories, which are particular cases of quantum theories. In the Schroedinger picture of quantum field theory the ordinary Schroedinger equation takes the form of a functional Schroedinger equation. Here states of a QFT are treated as functionals of the field coordinates in the same way as states in QM are treated as functions of the position coordinates. A thorough discussion of the the functional Schroedinger picture is in the article by Jackiw, Analysis on infinite dimensional manifolds: Schrodinger representation for quantized fields (p.78-143 of the linked document).

On the other hand, the Heisenberg picture is more general than the Schroedinger picture as it allows the discussion of time correlations of obsservable quantities. This is important in statistical mechanics, and essential for the study of quantum field theory at finite time, through the close time path (CTP) formalism. For the latter, see, e.g., Introduction to the nonequilibrium functional renormalization group by Berges.

As mentioned in the comments, the case of 4D relativistic QFT is somewhat peculiar since there are still unresolved foundational problems related to nonperturbative renomalization. As Dirac says, the ''operators'' in the Heisenberg picture don't act anymore on Fock space, so there is no Fock space Schroedinger picture. However, and Dirac does not say this, the Heisenberg picture gives (presumably, proved only in lower dimensions) a valid operator description in a different Hilbert space, and using the time generator of the corresponding unitary representation of the Poincare group, one gets a corresponding Schroedinger picture on this, renormalized, Hilbert space.

There is indeed a theory of differential operators applied on wavefunctions (or more accurately wavefunctionals), the Schrodinger functional formalism, though it is not used a lot in most applications. It is defined by a wavefunctional at a time t

\begin{equation} \Psi[\phi(\vec x), t] \end{equation}

Upon which act differential operators, defined by

\begin{eqnarray} \hat\phi \Psi[\phi(\vec x), t] &=& \phi \Psi[\phi(\vec x), t]\\ \hat\pi \Psi[\phi(\vec x), t] &=& -i\hbar\frac{\delta}{\delta\phi(x)}\Psi[\phi(\vec x), t] \end{eqnarray}