Turns out this is pretty annoying, as Amazon provides no API for doing this.

You can connect your Kindle to your computer and grab your My Clippings.txt.

You can go to https://read.amazon.com/notebook and scrape the page.

You can share the highlights directly from your kindle, which sends you an email with a link to a PDF containing the highlights. Which you can then scrape.

Or you can share the highlights from the mobile app, which also sends you an email, but with an HTML file containing the highlights.

There are various services around this, none of which I particularly want to pay for.

I'll probably just cobble something together with a PDF to text converter.