Can electrons travel faster than light in quantum field theory? How does quantum field theory account for the movement of an electron orbiting a proton from one place to another place in the next instant of time -
at a distance that even light couldn't reach in the time elapsed. The quantum mechanical wavefunction allows this.
 A: Electrons are not particles which orbit like planets. In the quantum field theory (QFT) that you ask about, they are distortions or excited states of the quantum field, somewhat akin to standing waves. In a hydrogen atom the electron state surrounds the proton as a bound state and does not change. The field equation describes the shape of this "orbital". It yields the probability of finding the electron at any given location, and so is sometimes referred to as a probability cloud.
Any "collapse of the wave function" due to measurement, as it is often described, is instantaneous across space. Even a photon from a distant star, whose wave stretches lightyears across space, will collapse instantaneously when it hits the astronomer's camera. But the collapse involves no "travelling" as such.
A: In relativistic quantum field theory (QFT) it is not the velocity that is fundamental but four-momentum:
$$p=\frac{mv}{\sqrt{1-v^2/c^2}}.$$
So all intermediate "trajectories" between two spacetime points (virtual trajectories, in the path integral formalism of QFT) do not have the quality of lightspeed greater than $c$. This speed is never exceeded. There is an infinite range of the vales of $p$ though.

How does quantum field theory account for the movement of an electron orbiting a proton from one place to another place in the next instant of time - at a distance that even light couldn't reach in the time elapsed?

It should be emphasized that QFT has difficulties in handling bound states. It allows us to calculate scattering amplitudes and decay rates with high precision. But in doing so external lines in the associated Feynmann diagrams refer to external lines that represent free fields. For an atom, no external lines are there. Hence the difficulty. Of course, in QFT no velocities $v$ which exceed the speed of light are present, as might be clear from what I wrote above.
You write:

How does quantum field theory account for the movement of an electron orbiting a proton from one place to another place in the next instant of time - at a distance that even light couldn't reach in the time elapsed.

First, there is no next instant of time. Instants have zero extent and putting two instants together will give the same instant. There is no next instant though it's obvious what you mean: to take the differential of time $dt$. They are defined as intervals with an extent approaching zero.
Secondly, in QFT an electron doesn't move in an orbit around a proton. So to say that in one $dt$ the electron is at a different position than the next $dt$ isn't
correct. Maybe this is the case in ordinary QM, but seeing QFT as lying beneath ordinary QM, this can't be the case.
The path integral in QFT stands in direct relation to the wavefunction in ordinary QM. But a thorough derivation of the wavefunction of an electron in interaction with a proton hasn't been derived, although the wavefunction can be derived in ordinary QM (but QFT holds the key to what's really "going on").
A: The quantity $v$ that appears in $\gamma = 1/\sqrt{1-v^2}$ is the group velocity of the wave function and it cannot exceed $c$. Hence no energy, charge or other physical quantity can travel faster than light in quantum mechanics.
To simplify notation I used units for which $c=1$.
