Posted from Voyager on Android.
Failed to load media :,(
Yes, it works just like any other image.

Some instances might not support uploading them. In that case you would have to host them somewhere else.
Edit: It also works with mp4 and webp.
Edit 2: Here, try it with this one:
https://swg-empire.de/pictrs/image/0d968fba-f087-46e7-9043-16d5bd178dc1.gif

gives you:
https://catbox.moe/ is great for gif hosting.
Same people have been having issues with catbox.moe, maybe https://postimages.org/ can be used too
Oh I get it now, you have to have the gif locally stored on your phone to upload it as an image with the image function.
If you want to embed a gif hosted by giphy its not as pretty.
If you have the direct gif url it should work fine.
Giphy works. You just gotta do it exactly like: 
On Voyager, I type the gif comment then tap the ••• button, “preview” to make sure it looks good before posting.
Giphy works. You just gotta do it exactly like:
Escape characters (\) are handy 
Is there a native button you’re selecting from within the app? When I click the image button on Voyager it looks like a normal upload file type function.
You forgot the exclamation mark.
What the above comment does plus just adding that in the [] brackets you can put a text description for the visually impaired if desired
Also if just don’t want to mess with formatting that some apps like voyager will do it for you by clicking the photo icon. That will upload the gif to your home instance rather than a 3rd party host site like catbox