Is it allowed for one to travel by car on Hoshana Rabbah? If so, what are the sources for this?
Hoshanah Rabbah is the final day of Chol Hamoed Sukkot. As such (to quote from this chabad.org article):
On the full-fledged festival days of Passover and Sukkot we are prohibited from creative work, much like Shabbat (with exceptions).
On Chol Hamoed, however, we are permitted to do many of these activities. For example, we may use electricity or drive a car.
However, as noted in this ou.org article:
What is important to keep in mind is that even the Melachos that are permitted on Chol Hamoed are only permitted if they are done l’tzorech hamoed (for the sake of Chol Hamoed, i.e. in order to benefit from them on Chol Hamoed).