Many hardware stores or glass shops will do the repair. There are however dozens and dozens of different types of storm window latches, some of which are obsolete. Hopefully the hardware store would have the type you need, or one that is close enough to use.

If you're going to do the work yourself, you'll have to risk breaking the glass, and there are two main ways that the latches are replaced. Some storm inserts are made in such a way that the latch easily slips out without taking apart the storm insert. Those types have a small hole in the bottom and part of the latch drops into the hole to interlock. By pressing the "button" that pokes through the hole it will pop out and is easily replaced.

The other type (the most common) is made so that the only way to remove and replace the broken latch is to take apart the storm insert. It's easier than you would think. Here's how:
The aluminum storm insert is normally held in place by 4 corner keys that are merely lightly crimped into place. Set the storm insert on a workbench. Take a chisel (I prefer about 1" wide), place it along the inside edge of the aluminum frame in the bottom corner (the chisel would be lightly touching the glass and is pressed against the aluminum, perpendicular to the glass) or into a weatherstrip groove near the corner and gently tap the chisel with a hammer to knock apart the aluminum extrusions in the corner. Once you have one corner pushed apart by about 1/8" you can start the other bottom corner. Continue to knock the bottom rail off, replace the broken latches, and then tap the rail back into place. Use a nail set to recrimp the corners. It seems as if you'd break the glass but that rarely happens (if done correctly). Again, you could have a hardware store do it (it actually only takes about 5 minutes) and let them worry about it. The following link shows about 50+ different types of latches:
http://www.blainewindow.com/catalog/itemlist.cfm?catid=193&compid=1&pcatid=74