Do atomic orbitals exist in a fully ionised atom? 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?
 A: 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.
A: 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

transitory phase

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) 
A: 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".
