Can you have an animated Steam avatar?

Yes, but only from the Points Shop. Steam doesn't let you upload your own. A GIF you upload as your avatar keeps just its first frame.

What you can upload

A still picture. Steam takes PNG and JPG avatars up to 1 MB and shows them at 184×184 on your profile. Anything bigger gets scaled down. Anything smaller gets scaled up and looks soft.

The Avatar Cropper crops any picture to exactly 184×184 and shows it at the smaller sizes Steam uses in friend lists, chat and comments. You can feed it a GIF, but what comes out is a still image, because that's all Steam will keep.

Getting an animated one

Animated avatars and avatar frames are sold in the Steam Points Shop. You pay with Steam Points, which you earn when you buy things on Steam (100 points per US$1).

Before you spend points, you can try avatars and frames on a mockup of your own profile in Steam Profile Studio. It shows the points cost of the whole look.

Crop a 184×184 avatar

Questions

Can I upload my own animated avatar to Steam?
No. If you upload a GIF as your avatar, Steam keeps only the first frame. Animated avatars only come from the Steam Points Shop.
How do I get an animated Steam avatar?
Buy one in the Points Shop with Steam Points. You earn points by buying things on Steam, at 100 points per US$1.
Does the Avatar Cropper make animated avatars?
No. It crops a picture to a still 184×184 PNG or JPG, which is what Steam accepts for uploaded avatars.