I wouldn't recommend using Tenor or Giphy unless you want your gifs to be used in every social media application and become public domain.
There are other file hosts you can use. (Though be warned that Imgur is blocked in the UK)
Viewing post in Gif's suddenly not animating
My solution is to put them in a public GitHub repo and link the raw content URL.
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.