does ''המצרכים'' derived from צריך ?? like 'what it needs' ?

"We don't have the soup ingredients" should be accepted

What if I said, "we don't have the ingredients to the soup"?

מצרכים

http://forvo.com/word/%D7%9E%D7%A6%D7%A8%D7%9B%D7%99%D7%9D/#he

I would say probably not. The direct object is used on ingredients making them "the ingredients". So the emphasis is on ingredients not soup. Then marak has a lamed joined to its beginning, which in this case becomes "for soup". It's a subtle difference, but it is a different sentence than you suggest.

How would you say, "We do not have the ingredients of the soup?"