Look at the part with the shiny collar threaded on one end - check out the hole at the other end from the collar. Is it threaded, and are those threads clearly part of the surrounding metal? If so, you've got a fixed thread size. Or, is there evidence that those threads are a separate part? If so, insert.
Sears used 3/4" - 16 threads on their lathes for years. This is a standard (fine thread) thread size/pitch, so if you can find a hardware store (or a bolt-and-steel store) with large bolts in both coarse and fine thread, take your chuck in and see if one will fit. The second most common size/pitch on American lathes is 1" - 8 (a standard coarse thread). If the 3/4" bolt slops around in the hole, try a 1" - 8 bolt.
The Sears lathes are kind of lightweight, so don't let your ownership of this chuck drive you in a decision to buy a lathe. You might be better off selling this one and using it to help fund a mid-weight lathe, like an older Delta. Those chucks are cool, but turners got along for generations without them.