Is there such a thing as lightweight window glass?
Well, there's rigid plastic, which is much lighter weight than glass for similar strength, but generally more subject to scratches. There are also plastic film products that would be similar to a screen in terms of weight and installation, some of which are very clear.

Glass weight is primarily a function of thickness, so thinner glass would be lighter than thicker glass, but without knowing what thickness glass you are contemplating now, it's unclear if you could go lighter or not.

Glass for a storm door should be tempered, which means 3mm / 1/8" is as thin as you can go (thinner glass is made, but it cannot be tempered and would present a cut hazard - since the tempering process also strengthens the glass, the thinner untempered glass would also be much weaker, and thus more prone to getting broken and causing cuts.)