since Saudi is a majority Muslim country and Islam is their main religion I can assure you there are plenty of proofs from the hadith of women not only traveling alone, ie without a males, but also of them having rode horses and camels.
What the case is here is that while certain scholars found women driving alone,without a mahram, impermissible, some namely like Sheikh Al Abani,Raheemullah, found it to be fine as long as it was for a good reason.
And the fitna in Saudi is that while a woman is not allowed to drive she is allowed to travel alone with an unrelated male driver which according to Islam is impermissible, Allahu alam. So all in all its a law that may have originally been established based on some wisdom at that time but is now been twisted and causing more evil,and God knows best.