does ''המצרכים'' derived from צריך ?? like 'what it needs' ?
Correct, same consonantal root.
"We don't have the soup ingredients" should be accepted
It sounds like mizrachim
ein lánu et ha-mitzrachím le-marák.
What if I said, "we don't have the ingredients to the soup"?
Doesn't sound right to me. Unless you can provide a context in which it works?
Haha true, true. At the moment it did.
"Have you made the soup?"
"No, I don't have the ingredients for the soup"
How would you say, "We do not have the ingredients of the soup?"
The english is incorrect. You cant say "ingredients for soup"
The English is fine for me as a native speaker. Its how I would always say it.