(What started as an attempt to write a simple answer has turned into a page-long ramble about the nature of quantum field theory and observables. However, I think the basic message is still there and worth reading.)
Background: scalar fields and charge conservation
We can actually uncover the salient features of how to properly define and interpret the "charge" of a quantum field by just considering a complex scalar field theory. The Lagrangian for such a theory is
$$\mathcal{L}=\partial_{\mu}\phi^*\partial^{\mu}\phi+\cdots,$$
where the extra terms are possible interactions or mass terms which are irrelevant for our discussion. This theory contains a global $U(1)$ symmetry $\phi\to e^{i\alpha}\phi$, $\phi^*\to e^{-i\alpha}\phi^*$. The conserved current for this symmetry is easily obtained from the Noether procedure as
$$j^{\mu}=-i\left(\phi^*\partial^{\mu}\phi-\phi\,\partial^{\mu}\phi^*\right).$$
As we know from classical field theory, the charge density associated with this global symmetry is given by $j^0$, and the total conserved charge can be expressed as a functional in the field variables as
$$Q[\phi]\equiv-i\int\mathrm{d}\textbf{x}\left(\phi^*\dot{\phi}-\phi\,\dot{\phi}^*\right).$$
Once an electromagnetic field is turned on, this turns out to be the electric charge associated to the electromagnetic interaction. In this sense, your question is very easy to answer classically:
The charge associated to a classical field is spread everywhere that field has support, as a classical field carries a charge density $j^0$ associated to its Noether current.
Quantum field ontology
So what does this have to do with particles and other nonsense? How can we move from the interpretation of a classical field as carrying a classical charge density to that of a particle carrying a fundamental charge?
In order to answer this question, it's important to remember what we're doing when we do field theory. We have to remember the following:
Quantum field theory is not the study of fundamental particles, but, rather, the study of the quantum mechanical analogues of classical field theories.
That fundamental particles arise in the study of quantum field theory should, for the sake of this argument, be seen as a happy historical accident. What is most important is that we study the quantization of classical field theories.
To quantize a classical field theory is very simple, but somewhat subtle. We continue the same way that we quantize classical mechanical particles. We identify our canonical variables and associate to those variables a wavefunction that obeys a Schrodinger equation. In field theory, the canonical variable is $\phi(\textbf{x})$ -- the field configuration at any given point in time is the observable variable. Thus, it is natural to define a wave-functional $\Psi[\phi]$ which associates to each field configuration a quantum-mechanical amplitude for that configuration.
After we introduce the wave-functional, we wish to solve the Schrodinger equation
$$i\partial_t\Psi[\phi]=\hat{H}\Psi[\phi].$$
We do this the same way we do in standard quantum mechanics -- we break up $\Psi$ into a sum of energy eigenfunctions, then let the diagonalization take care of the time evolution.
We can (and should) also introduce a Dirac-type notation where $\Psi[\phi]=\langle\phi|\Psi\rangle$, where $|\phi\rangle$ is a state of definite field configuration (that is, a state such that a measurement of $\phi(\textbf{x})$ at any point will yield a definite result). This helps to clean up the notation.
Also, in direct analogy to the definition of momentum in standard quantum mechanics, we associate the conjugate momentum of $\phi$ (in our case, $\dot{\phi}$) with the field-momentum operator, which takes the following form when acting on wave-functionals:
$$\hat{\pi}(\textbf{x})=-i\hbar\frac{\delta}{\delta\phi(\textbf{x})}.$$
Thus, the operator corresponding to the electric charge is given by
$$\hat{Q}=\int\mathrm{d}\textbf{x}\left(\phi^{\dagger}\pi-\pi^{\dagger}\phi\right).$$
Back to defining charges
Let us say that we have diagonalized the Hamiltonian of our particular theory. We know that since $Q$ is conserved, when it is promoted to an operator, we will have $[\hat{H},\hat{Q}]=0$. Thus, the Hamiltonian and electric charge can be simultaneously diagonalized. That is, we can choose an energy eigenbasis such that each basis wave-functional will have a definite energy and electric charge.
Let's say that $|E,q\rangle$ is some state such that
$$\hat{H}|E,q\rangle=E|E,q\rangle,\hspace{0.5cm}\hat{Q}|E,q\rangle=q|E,q\rangle,$$
for some energy $E$ and some charge $q$. Where would you say that the charge $q$ is located? This is not a question that has an answer in the quantum case, as the state $|E,q\rangle$ is in general a specific superposition of states of different charge density distributions. The only thing one can say is that, once you measure $q$, you know it must be conserved in time.
Asking where the charge is located in a specific state of a quantum field theory is sort of analogous to asking where the electron is located in an electronic orbital (ie, asking the value of an observable in a state which is not an eigenstate of that observable). It isn't quite right to say that the electron is "everywhere", because that's not how superpositions work. Indeed, the question of "where" only makes sense upon measurement.
In the same sense, the question of what the charge density at a certain point in a general state in a QFT is constitutes an equally nonsensical question. It is only when one observes the charge density at that point does the question make sense.
The question of "What is the charge density configuration?" for a given state in a QFT is just as nonsensical as asking the position of a particle in standard quantum mechanics. It only makes sense upon observation.
Particles (finally)
So far I haven't made any mention of how particles fit into all of this. The reason is that "particles" are simply special states in translation-invariant field theories. Namely, a "particle" is a state of definite energy, momentum, and charge (others will have a definition of "particle" in QFT that differs from mine, but the argument remains the same). Whether we're talking about particles or more generic states, the conclusion above remains the same. Asking about the charge density distribution in QFT is an ontologically nonsensical question until measurement.
I hope this helped! It's certainly confusing and hard to think about these things, and perhaps even harder to write about them. So if anything is confusing or just wrong, feel free to ask/correct me!
