Nonsense. If a state legislature passes a law that is unconstitutional (whether it be the state or federal constitution) because it is too vague, it too can be struck down by the courts.
Arbitrary and capricious is not the same thing as vague. E.g. a law that says you can't drive a yellow car on the second Friday of each month is arbitrary and capricious but it is not vague (except maybe in the definition of "yellow")
The "arbitrary and capricious" language is used specifically in the Administrative Procedural Act, and so you don't see it much when talking about state law, but state laws have also been struck down for being arbitrary and thus in violation of the Due Process clause. See e.g. https://law.justia.com/constitution/us/state-laws-held-uncon... (search for "arbitrary").
See e.g. https://www.law.cornell.edu/supremecourt/text/405/156