After a little more testing, it turns out IndexedDB might not actually be supported in Safari.
Safari does not support IndexedDB for content running in an iFrame.
https://docs.unity3d.com/2019.1/Documentation/Manual/webgl-browsercompatibility....
I was seeing it work at first, however when I close the Safari app the data is no longer persisted.
I'm not sure if it's possible to embed games on itch without using an iframe?