Most early wire trivets were individually hand crafted by tinkers- men who worked with tinned wire. Books on Victorian Wireware show early photographs of these men working with bolts of wire and pliers. Wireware items were also made by peddlers or gypsies.
A number of catalogs from the late 1800s contain wireware. In addition to trivets, there were elaborate serving pieces and other decorative household accessories fashioned from wire. Some simple pieces with plain designs could have been factory made.
People lost interest in wireware in the early 1900s and it pretty much went out of production. Of course there are still some limited wireware items available today, but most of these are fashioned of stainless steel or aluminum wire.