From the rigorous point of view, the observable vacuum sector of a relativistic quantum field theory (QFT) on flat Minkowski space is defined by the Wightman axioms. (There are also variations of these in terms of nets of local algebras, but the Wightman axioms are considered most basic; they are also the criterion to be met for a solution of the Clay Millenium problem to construct a QFT for Yang-Mills. There you can also see how the vacuum sector of a gauge theory fits in conceptually. The unsolved conceptual problems that you allude to concern the charged sectors only.)

Given the Wightman axioms, the observables (in the sense of potentially measurable operators) are the smeared fields obtained by integrating the distribution-operator valued fields with an arbitrary Schwartz test function, their products, the linear combinations of these, and their weak limits, as far as they exist.

The state vectors are the products $\psi=A|0\rangle$ where $A$ is an observable and $|0\rangle$ is the vacuum state. (Of course, many different $A$ produce the same $\psi$; e.g., for a free QFT, one can change $A$ by adding any operator of the form $Ba(f)$ where $a(f)$ is a smeared annihilation operator, without changing the state.)

The dynamics is dependent on the choice of a time direction along positive multiples of a timelike vector $v$, and is given by $\psi(t):=A(t)|0\rangle$, where $A(t)$ is obtained from $A$ by replacing all arguments $x$ of field operators in the expression defining $A$ by $x-tv$. The latter operation is an algebra automorphism believed to be always inner, i.e., induced by conjugation with a strongly continuous 1-parameter group generated by a $v$-dependent Hamiltonian $H$ with $H|0\rangle=0$. Assuming this, the Schroedinger equation holds.

To get a more concrete view of the Hilbert space and the dynamics one must either consider exactly solvable QFTs (of which nontrivial examples currently are known only in spacetime dimensions $<4$, and indeed, in 2-dimensional conformal field theory one can give a much more specific picture.), or sacrifice rigor and consider renormalized perturbation theory. In 4 dimensions, the latter builds the Hilbert space as a formal deformation of a Fock space and the fields as formal power series in $\hbar$ or a renormalized coupling constant, although to get physical results one hopes that these formal power series can be evaluated numerically by appropriate trickery. In case of QED this works exceedingly well, but less so in other QFTs.

Alternatively, one discretizes the QFT on a finite lattice, and reduces the problem in this way to one of ordinary quantum mechanics, hoping that for a fine enough and large enough lattice, the results close to the continuum results.

One can also use the functional Schroedinger representation, though this is not mathematically well-defined. Note that contrary to the false claim unlike the functional field equation discussed in the article cited by the OP (which is a philosophical, not a physics paper), the functional Schroedinger equation is in general not equivalent to the Fock representation. In particular, unlike the Fock representation, the functional Schroedinger equation is able to explain many nonperturbative features of interacting QFT. See the discussion of Jackiw's work.

For nonrelativistic QFTs, the situation is somewhat simpler, as particle number is conserved. In the vacuum representation, the Hilbert space is a proper Fock space, and splits into a direct sum of $N$-particle spaces to which standard quantum mechanics applies. However other representations such as those relevant for equilibrium thermodynamics, some of the problems from the relativistic case recur, since the appropriate Hilbert space is no longer a Fock space.

In curved space, no good system of axioms is known, and one generally uses a Fock space perturbation approach with all its limitations.