Univega Alpina 503
Retro Newbie
If it is not clicking, it is because the ratchet pawl is stuck due to old grease. There are two. Douse with degreaser or WD-40 if you do not have any degreaser, let it sit for a few minutes, then work the pawls until they move freely. They have tiny springs of a similar size to what you might find in a ballpoint pen. If you can, take everything apart, but be careful, these are easy to damage, clean and lubricate and put back together. The plastic broke, taking mine apart.
Option two is to replace them with something inexpensive on AliExpress. The only problem here is that most of the combined Shimano brake levers/shifters they sell are made for V-brakes. Cantilever brakes require less wire travel but more effort, so they are linked differently at the brake handle. In a pinch, you can use V brake levers, but they will require more effort to pull than your brakes do now.
I would recommend replacing them with separate brake levers and shifters rather than integrated ones, as it will make it easier to get cantilever levers and shifters with the appropriate gear range.
Option two is to replace them with something inexpensive on AliExpress. The only problem here is that most of the combined Shimano brake levers/shifters they sell are made for V-brakes. Cantilever brakes require less wire travel but more effort, so they are linked differently at the brake handle. In a pinch, you can use V brake levers, but they will require more effort to pull than your brakes do now.
I would recommend replacing them with separate brake levers and shifters rather than integrated ones, as it will make it easier to get cantilever levers and shifters with the appropriate gear range.