are you allowed to drive your own vehicle through zion national park in late april early may?

Yes. It's accessible.

That's a great question! I believe March through November are National park shuttlebus only. With the government shutdown(s) it had gotten complicated. I was recently there and was able to drive though. You can find more current info on nps.gov

I don't really know but I'm sure you can call the rangers station and find out

Most of the park is accessible all year , there is also a shuttle to take you where vehicles cannot