Is there a 32-bit build for Linux, or is it possible to use the source code to build my own?
Most 32-bit Linux distributions do provide a Tiled package in their repositories, but of course it may be a little out of date. Alternatively, there's a Tiled snap package that includes an i386 build. You can definitely also compile Tiled yourself, the source code is at https://github.com/mapeditor/tiled and there are compile instructions in the README.md.
Thank you very much for your help.
Although what I was hoping for was some sort of portable install file I could copy onto a slew of 32-bit Linux boxes (such as a .deb file) from a Flash drive.