This isn't all that hypothetical. The specific example is, but new colonies almost always need things from "civilization" that they can't produce locally, because they haven't yet built up the economic base. Few nations today are totally self-sufficient. Every country in the world imports something. How dependant they are on that import varies. Take it down to a low-enough level, like one city, and no city is completely self-sufficient.
The people supplying a vital resource may think, "Hey, we really have these folks over a barrel. We can charge outrageous prices, force them to support us politically, etc." But in real life, usually not. If the price -- whether that price is cash or some other concession -- gets high enough, the buyer will usually find an alternative source. Either somebody else who can also supply the same product, or some similar product that can satisfy the requirement.
Like during World War 2, the Allies blocked shipments of rubber to Germany. So the Germans found ways to recycle old rubber products and to produce synthetic rubber out of other materials.
In the 1970s, the Arab oil-producing countries used oil as a political weapon against the West. The West responded by finding new sources of oil, such as the North Sea and northern Alaska, increasing energy efficiency to reduce the demand, and to some extent tinkering with nuclear, solar, and other sources.
You can write a hypothetical situation on a piece of paper, "This resource is vital to human survival, it is impossible to produce locally, and there are no substitutes." But in real life such a situation rarely, if ever, happens. It may be very difficult and expensive to produce locally, but it's usually not impossible. If the suppliers demands are great enough, you can bit the bullet and do it. There is almost always some substitute. If we can't get peanuts, we can get walnuts. If we can't get oil, we can burn coal. Etc.
Developing an alternative can take time and effort, so the supplier may be able to inflict short-term pain. But unless they can literally destroy the society, their gains are as short-term as the pain. An embargo is most effective in war time: If you cut off some crucial resource, you can reduce the enemy's ability to fight, and then defeat them before they have time to develop an alternative.