What ThinkPad/Latitude models are cheap and usable right now? I'm just looking for something to browse the web, but not something so low-spec that it couldn't handle heavier desktop environments.
Define cheap. The least expensive laptop on Dell Refurbished currently is $180 and would easily run any desktop environment, including the heavyweights. Specs are here:
CPU
1x Intel Core i5-6300U (2-Core, 2.40 GHz)
Memory
8 GB (1x 8GB)
HDD
256 GB (1x 256 GB SSD)
Display
14" HD (1366 x 768)
If you're thinking cheaper yet, you'll want at least a dual core processor and 4GB of RAM. Just about any business laptop from the last 10 years or so would work, as long as you stay away from bottom of the barrel Celerons or AMD processors and <4GB of RAM. You can run Linux on a very low spec machine, but you'd want to use a lightweight DE and web browsing wouldn't be a fun experience.