What is the ruling on private car owners exploiting the need of travelers during Eids and occasions by transporting them for a doubled fare, and is this considered a violation of Islamic law due to the absence of a transportation license from the state?

The laws imposed by state systems are binding if they serve a clear and public interest, and if violating them leads to a clear and public harm. However, if the interest is not clear, then one is bound by it outwardly only, and not inwardly, especially in times of need, as stated by the Shafi'is. As for excessive transportation fares, they are not forbidden if both parties consent, but it is disliked to exploit people's needs and overburden them. Indeed, the Prophet, peace and blessings be upon him, prayed for mercy upon the lenient merchant.