I know that vegetable is a count noun, but Which one of the quantifiers do we use with vegetables. How much vegetables do our bodies need? How many vegetables do our bodies need?
From Longman Dictionary of Contemporary English:
Grammar Vegetable is a countable noun, not an uncountable noun: They grew their own vegetables (NOT their own vegetable).
Using the word in plural form basically refers to all the various types of vegetables; i. e. carrots, tomatoes, parsley, etc.
However, talking about the quantity, it'll depend on the type of the vegetable you're talking about and the general context.
You'll ask a quantity-related question about vegetables if you 're buying some; then if you're buying it by weight, that'll be for example two kilos of vegetables, so the question would be "How much vegetables ...?"
Although, that'll apply to the countries with good climate for growing vegetable, where in the other parts of the world, specially in the West, where shops don't usually sell vegetables per weight, they'll come in packs or pots. So the question would be "how many packs/pots of vegetable...?"
Peter Shor answered this nicely in a comment:
You should use "how much", because you are not expecting an answer of the form: "our bodies need thirty-seven vegetables". However, "vegetables" is unusual in this regard. Usually you wouldn't use the plural so you'd say: "how much apple does the recipe call for"? But I think you have to use the plural for "vegetables", because "vegetable" is a mass noun with a different meaning (In the broadest sense, all plant life and plant products.)
Usually, it should be "how many vegetables". For example as in "how many fruits and vegetables should I eat a day?"
Logic behind it
It depends on the context and the answer that you expect. Thus, the questions should be answered only in combination with a specific context. If you expect a certain integer number of items (vegetables), then clearly you are asking for the number of a countable noun. If instead, you are asking for an amount measured in physical units, then the noun is usually uncountable (water) and requires asking for "how much [water]". In some cases, the same noun can require different pronouns due to different usages.
Try searching "how much vegetables per day" with Google. Google suggests instead that you search for "how many vegetables per day" and only displays such search results since usages of the noun vegetables in which it is considered a countable noun are more common.
If the question is simply "how many vegetables", then you are asking for a specific number of vegetables, and "many" is the correct word. If you aren't expecting a number of vegetables, you should modify it with how many "cups" or "pounds" or other unit of measurement.
In general, 'how much' refers to a quantity of units (money/a lot or little, water/some or 2 liters, exercise/occasional or 30 minutes). It is for more nebulous answers, such as some or a lot or none. Answers can be in general quantities, or sometimes even in specific units, but the item itself is not countable, only the units to measure it.
'How many' is for countable items. How many cups of water? How many vegetables? How many dollars? The answer is usually a number, although 'a few' or 'some of them' may work as well.
How much chopped bell pepper is acceptable (2 cups), whereas how many chopped bell pepper is not. How many chopped bell peppers is acceptable (2), as is how many bell peppers do I chop. How much bell peppers do I chop is not ok, although how much chopped bell pepper is. The former is refering to the bell peppers and requires a number answer refering directly to the bell peppers, the later refers to a unit (cups, grams, hogsheads, etc.) of mesurement containing chopped bell peppers.
TL;DR - If your answer is a number, it is probably how many. If your answer is a number of units or a vague quantity, it is probably how much. Since it is English, there are of course exceptions.