Is your storm door glazed with an approved safety glazing material? If so, there will be a permanent marking indicating the company that made the glazing, the type of material, the minimum thickness and, most important, the legend "ANSIZ97." That tells you that the glazing complies with American National Standards Institute standards, a legal requirement in many states.
If the glazing doesn't have that Z97, you should replace it with glazing that does meet Z97 before the old glass breaks and injures someone.
You can use tempered safety glass, one of the super-tough plastics such as polycarbonate (Lexan) or acrylic (Plexiglas). Lexan is the strongest (though all are strong enough), but it is not as clear and ripple free as the others. Plexiglas is almost as strong but, like Lexan, can be easily scratched. Optically it's as good as tempered safety glass, which is the easiest to keep clean and scratch-free. Balance these factors against price and waiting time (the glass may require a special order).
Incidentally, save the old glass to repair broken windows. It's fine for use in locations less hazardous than storm doors.
Generally, the best material to use is 3/16-inch acrylic plastic. It's easy to work with, cheaper than polyearbonate, and comes with installing clips that eliminate the frame used for the old glazing. That simplifies and speeds your task.
To get the right size, remove the old glass and frame. Measure the opening and subtract for thermal expansion: 1/16-inch for a pane less than 36 inches longest measure; 1/8-inch, 37 to 48 inches; 3/16th-inch over 48 inches.
The plastic may be cut to size at the shop. But if you do the job yourself, pick up a scriber-type cutting tool when you buy the plastic. Draw it over the plastic about 10 times, using a straightedge for a guide. Then place the plastic, scribed side up, over a 3/4-inch dowel (or any similar-sized rod, like a curtain rod). Line the dowel up under the scribe line and press the plastic down over the dowel to break it along the line.
Sand the edges of the plastic with 80-grit sandpaper wrapped around a block, just enough to smooth the edges and to round them slightly.
Now, remove the protective paper from the plastic. Place it, unframed, into the door opening, and tighten the retaining clips you've bought with the plastic. If the glazing is too thin for the clipsto grip, slip thin wooden spacers between clip and plastic.