Using Civi to manage members' lists of items available for donation to other members
I also received a helpful answer from Rose in the Civi chat (thank you Rose!). Thought I'd post it here just to have it all in one place: 

Interesting problem, and there's probably a way somehow. I've also looked at the response in Stack Exchange, which definitely recommends Drupal because then you can use Views and Webforms. A few questions first... 1) Why does it have to be Wordpress? 2) Do the schools need to log in/record their own needs and surplus or will it all be managed by staff in the organisation? 3) Are the items specifically limited (e.g. 3 tables, 100 books)?

A really simple method would be to just use custom fields to record what items are in surplus and then use Reports to pull off which items are needed, and use customised Activities to record any transactions between schools. Drupal is probably key if you do want the schools to log in and record their surplus or needs. You could use Memberships and Donations (of a sort) to record items donated but it's not as simple to record how the items are transferred to the under-funded schools. Possibly Grants but it's not a very sophisticated tool. 

And Drupal is probably key if you want to display the info online (using Views) although you may be able to use the API to display some information but you can't interact with the information using the API. For example, people with some log in access can view information and then submit a request for it if you're using Drupal.

Sorry -  discount my second question because you have already answered that! I asked the third question because if 'yes' then the answer in Stack Exchange is probably a better answer than mine, but if they're happy with using custom fields to just record categories of items (plus a free text field to provide a bit of detail, including condition), this would be sufficient for schools to see what's available.