if you click on the ground your sim should have the option to start a protest ... if they don't then it will be because protests are tied to the political career. I thought that they were not.
Anyway, once your sim has chosen a cause (click on sim) then they can start a protest. Once protesting click on another sim and ask them to join (under friendly). The make your sim quit their protest and now you can donate to the protesting sim.
Now, my sim is currently in the Political career so this may only be, like you say, for sims in that career. If that is the case you have 3 options, cancel the whim or make your sim join the political career... or add a random sim to your household, make them join the political career and start a protest, fulfill your original sims whim and then move out the recently added sim.