As per @izzy's suggestion in the comments, I've now found a number of theoretical solutions, and one that works, enough, in a way that I need.

I managed to get the Traccar app send data to a Traccar server, but I do not see a straightforward way to then get data out of the Traccar server.

There's also php-tracks-recorder, a web-based package which is supposed to be able to receive payloads from Owntracks and Overland, both mobile apps available on iOS. At the moment, I only get Owntracks to push one point to php-tracks-recorder. But, Overland happily chugs away, pushing continuous updates to php-tracks-recorder, which dutifully records the data.

The above is not ideal, but should solve my most pressing need: background location tracking, pushed to the web.

I would still need to write code that allows me to export the locations from php-tracks-recorder via, say, JSON.

This is not an ideal solution for tracking multiple devices, as device identification depends on the user setting the device ID inside the tracker app, meaning multiple users easily can have the same ID.