Yvette - they will need to get an NIE number to buy a car. I don't know whether they will be able to get one if they are just on holiday - although I suppose they must be able to as many home owners aren't resident either. Not difficult to get - just tiome consuming. Ours took 4 weeks to come through - and that was done urgently.
We also had to produce our Padron registration - and that is a bit more tricky as this is what proves you have a property in Spain and you have to produce your escritura to get this. . . . Although I do have friends that weren't asked for their padron when buying a car.
We bought our car using our UK license - don't think you need a Spanish one.
May need to have a Spanish bank account to make payment from - we certainly were asked for our bank account details so that road tax could be taken automatically each year.
Insuring it might also be tricky. Ditto for paying the Road License - as the way things work in Spain is that whoever owns it in a particulr month has to pay for the full year (can't remember when the month is - February I think).
They will have to use an address to buy it from and register it and insure it at. I would be unhappy about someone using my apartment address - but I'm just paranoid. I see all sorts of problems that will proabably never happen! However . . .
Some friends of ours have had major problems with the car they sold over 18 months ago. They sold it to a garage and it is still in their showroom but our friends keep getting demands to pay the road tax and get the ITV done (MOT) as the garage hasn't actually changed the registered owner. One friend I talk to on another forum got done by the police for 15 years back road tax as for whatever reason the re-registration didn't happen properly when they sold it.
Hugh-Man may be able to give you more info as he's done it more recently than I have in Murcia (I'm in Alicante) so may bde able