Say a H atom is ionised and then it captures a free electron at a later time, do the atomic orbitals then have to go through a transitory phase to accommodate the electron before they form the correct wave function?
Adding an electron to a bare $H^+$ ion is an important process in astrophysics.
It is also a relatively low probability process because energy must be released by photon emission (or some other process) to enable the electron to drop in energy into an Atomic Orbital.
I agree with ACuriousMind that whether the orbital exists or not when unoccupied is a matter of definition.
I would like to add that there is not any
An electrons drops into unoccupied orbital level and a photon is released.
This process is known as radiative recombination. A description is given here.
Edit after good comment from Floris. If the electron only remains in the atomic orbital for a very short time after it is captured then energy/lifetime uncertainty broadening means that the energy of the atomic orbital will be a bit uncertain and a somewhat broad range of photon energies would be observed. If, on the other hand, the state formed is stable on a longer time scale the photon released would have a narrower energy range and the energy of the atomic orbital would be better defined. (Final comment - the energy of the photon will be the energy of the electron intially plus the energy release in formation of the hydrogen atom - if the free electrons which are captured by the ions have all have close to zero kinetic energy - or all have the same energy above zero then we might expect the photons emitted to give a spectrum of lines reflecting the energy lost by the electron as it is captured)
Atomic orbitals are states of electrons. If there's no electron, then all orbitals are empty, since there's no electron state occupied. Whether a space of states that is fully unoccupied "exists" is an irrelevant matter of definition.
Atomic orbitals are the states of the mutual motion! Few people know that the nucleus moves around the atomic center of mass and produces a "positive charge cloud" or "orbital" strongly depending on the state $\psi_n$. If you consider a positronium, both "clouds" coincide in size. When we speak of an orbital, it is normally the relative distance who is an "orbital".