I have a partial solution:
Store your animated GIFs in a public GitHub repository and link the raw content URL in the HTML/Markdown with an <img> tag (HTML) or a ![]() (Markdown).
Here’s an example:

- Image link: https://raw.githubusercontent.com/jbunke/tdsm-art/refs/heads/master/_tdsm/logo/itch/tdsm-itch-card-3-mini.gif
- Repository: jbunke/tdsm-art
This works for GIFs in project page and devlog bodies, but it doesn’t work for banners or background images, since those have to be uploaded via the theme editor, and thus they rely on Itch.io to be animated and rendered correctly.