The noun "noodle" is countable, and this expression is grammatical:
I ate many noodles today.
However, it is unidiomatic. People don't count the number of noodles they are eating. (Unless they are fanatical dieters!)
The idiomatic expression is
I ate a lot of noodles today.
The situation with "vegetables" is a little different, because "many vegetables" can mean a wide variety of different kinds of vegetables.
You should eat many vegetables.
probably means that you should eat many different vegetables.
If you are saying that someone should increase the amount of vegetables in their diet, then you would say
You should eat a lot of vegetables.