Can I add body to vegan stock with carrageenan?
Carrageenan is not suitable for the application you propose.
First, you need iota carrageenan, since the kappa variety is inhibited by salt.
Second, you need a source of calcium ions for the iota carrageenan to form a gel (kappa needs potassium). I doubt that it will be sufficient to chuck a piece of kale in there, you will probably need something more. I don't know for sure how it is done in vegan cheeses, but this type of preparation is frequently firmed up with inorganic salts. If you were to add calcium sulfate to a stock, you would end up with something rather unpleasant, I imagine.
Third, the carrageenan action is also inhibited by long heating at low pH. A stock is usually not too low in pH, but it is lower than neutral.
In short, my suggestion is to pick a better suited thickener.