Are modern scientific discoveries of importance ever made by amateurs (as opposed to institutions)?
That depends on what you mean by “modern”. Darwin was an amateur and look at the body of work he turned out. Not all of us are as rich as the Darwin-Wedgewood line and, as you noted, financial backing is hard to come by if you are an amateur. Still, it is theoretically possible. Most advances are based on previously-published material and some of it comes from amateurs.
