You can add GIFs to your signatures and most platforms will deal with them just fine. The exception is Outlook (all versions with the exception of Outlook mobile apps and Outlook.com webmail) which doesn't play the GIF but only displays the first frame as a static image.
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That doesn't mean you shouldn't use GIFs - just be careful and make sure that all the important information is contained in the first frame in order to deliver your message even to Outlook users.