In the path integral for quantum field theory, do the intermediate states have quantized particles? BACKGROUND
I have been following chapter 9 of Peskin and Schroeder where in section 9.2 the functional integral formalism is used for the quantum theory of a real scalar field.
As far as I understand, the path integral operates over all intermediate field configurations from a specified start state to a specified end state.
In particular, there does not seem to be any constraint that the intermediate field configurations are quantized in any way, e.g. to have integral number of particles.
QUESTION
Are the intermediate field configurations constrained to have an integral number of particles?
If so, then where does this appear in the mathematics?
If not, then why does the physics not include terms where fractions of a particle appear?
 A: When the path integral is used to define a quantum theory, the intermediate states are classical, so the question doesn't make sense. 
You can see this in the path integral for quantum mechanics. In the usual formulation, we consider paths $x(t)$. Each path describes the motion of a classical particle; the quantumness comes from integrating over paths, not the paths themselves. Similarly, in the path integral for quantum field theory, the intermediate states are simply classical field configurations $\phi(x)$. The number of particles in a field is a quantum notion and does not apply.
But there's a closely related question that I think you wanted to ask: if we want to use the path integral to find the amplitude to go from one quantum state to another, how do we map the quantum states to classical boundary conditions? 
For quantum mechanics we associate a classical particle with position $x$ with the quantum state $|x \rangle$. In quantum field theory we usually associate a classical field $\phi(x)$ with the field eigenstate $| \phi(x) \rangle$ which satisfies
$$\hat{\phi}(x) | \phi(x) \rangle = \phi(x) |\phi(x) \rangle.$$
Since $\hat{\phi}(x)$ contains creation and annihilation operators, $|phi(x) \rangle$ can't have a definite number of particles, or even a bounded number (because if the highest number was $n$, acting with $\hat{\phi}(x)$ would give a part with $n+1$ particles). Instead, it's a superposition of all possible numbers of particles. This is pretty unwieldy for scattering problems, but as you've seen in P&S, we can compute $S$-matrix elements without thinking about the boundary conditions at all, which is why they spend so little time on the subject. 

A third point is that, instead of using the path integral to define the quantum theory, you may instead want to start in the quantum theory and derive the path integral. In quantum mechanics, we do this by inserting resolutions of the identity in the position basis, 
$$1 = \int\, dx |x \rangle \langle x |.$$
For quantum field theory we instead use the field basis defined above,
$$1 = \int\, \mathcal{D}\phi(x) \, |\phi(x)\rangle \langle \phi(x)|.$$
In this picture, the intermediate states are quantum. They don't contain a definite number of particles, but instead are superpositions of states that are. 
I totally understand being confused about this, because P&S shove a whole lot under the rug. They run through everything I said in about one paragraph, leaving everything implicit, and elide the difference between the path integral as a definition of a quantum theory (where the intermediate states are classical) and the path integral as a derived consequence of a quantum theory (where the intermediate states are quantum), switching between the two in a single sentence. For more detail about path integrals, I recommend these lecture notes.
