First, create a GIF file of your icon

After that, go to the website - https://ezgif.com/

https://steamuserimages-a.akamaihd.net/ugc/2518156842401630717/B3E73A1B4F5782BF37DAC559823A3D74A6452D13/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false

https://steamuserimages-a.akamaihd.net/ugc/2518156842401631246/CC1A7B24C2419EB350AE8A0A3646B7A1E8449F69/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false

https://steamuserimages-a.akamaihd.net/ugc/2518156842401631861/EDB81D2C357660C529124E6E6B8B321F9DB02B2E/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false

Upload an gif and click split into separate images

Then download the archive

Now you have all the individual images of your gif file, I hope you are able to create an animated texture

In VTFEdit, import all the images by highlighting them

Save the texture, then set up the VMT file
Write it down:

Proxies
{
AnimatedTexture
{
animatedTextureVar $basetexture
animatedTextureFrameNumVar $frame
animatedTextureFrameRate 25
}
}

https://steamuserimages-a.akamaihd.net/ugc/2518156842401661766/8B8C03F08F7E47A0E30C6CB9E14C8C3341835275/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false

https://steamuserimages-a.akamaihd.net/ugc/2518156842401662554/86278C745CA4827EB710880DD235DEA60DD2F106/?imw=256&&ima=fit&impolicy=Letterbox&imcolor=%23000000&letterbox=false

It turns out everything, the texture should work