No, you can’t have a Slack profile picture gif.

I’m sorry but, No.

You can’t have a Slack profile picture gif. 

At least for now.
This was even confirmed by the Slack team in Feb 2018 on Twitter.

Be sure that I would have one for myself if this was allowed.
As I already shared how much I like custom animated gif on Slack.

Your profile picture on Slack needs to be an image with a minimum of 512×512 pixels, but no larger than 1024×1024 pixels.
Here is the Slack help page about the profile picture.

Though, if this is really important for you to have it. You might consider switching to Discord, as they allow it.