As I stated below Car tax is collected via the Town Hall and for this you need a Padron.
Yes you can buy a car without the Padron and it may be taxed when you buy it for a period.
If you have a car and no Padron then you could be driving without Tax.
I have discussed the issue of Padron' s with the Mayor of Alhama and our Admisistration, and the Mayor stated they will only be issued to people who spend 6 months a year in the area. (His words) (how this is proved or checked is another matter) This came about when car hire prices made it viable to own one, but people at Condado were being refused Padrons by Alhama Town Hall.
Some councile will give you one without question.
At the end of the day if you can get the Padron then good if not you risk Prosecution for Driving without Tax.