What type of brake is this?
As stated by Daniel R Hicks, these are roller cam brakes.

Sheldon Brown sells a few extra long reach brakes: the Odyssey 1999 Extra Long Reach Caliper Brake and the Action© Extra Long Reach Caliper Brake. Tektro C-326 seems to have adequate reach, but the problem is since its designed for beach cruisers, the rims have to be quite wide. I don't know if the other brakes listed have this problem.

A relatively easy solution is a drop bolt, which is a device which allows you to mount a brake lower than the frame is designed to. You can make a drop bolt and use a regular reach brake instead of hunting down a 100 mm reach brake. These are reasonably common with the Raleigh Twenty crowd, for example. In the picture below from Sheldon Brown, he has a Raleigh Twenty with a short reach caliper!



(Image from Sheldon Brown)

Note that this is a hack, so its YMMV/at your own risk.

A real drop bolt looks like this: