When it comes to sending animated GIFs, each email client that supports HTML format will do. However, if you send your message in plain text, it will not do.
When it comes to displaying animated GIFs, it gets complicated. All web-based email clients (like OWA, Gmail, or Yahoo) should have no issues displaying animated GIFs. In most cases, Outlook for mobile and native Android & iPhone mail apps show animated GIFs typically.
Desktop versions of Outlook, in general, don’t support animations in emails. “Lack of support” means that GIFs appear but are not animated or appear as flat images. Instead, those clients will display a static image of the GIF’s first frame. The only Outlooks in which animated gifs have been seen to work are Outlook 2003 and Outlook ProPlus (Outlook as a part of Office/Microsoft 365 subscription), according to official release notes, starting from version 1901 of the monthly update channel. The semi-annual update channel has supported animated GIFs since version 1908.
The information above hardly counts as the complete list of supported email clients. But it’s not the point. The point is that the animated GIF might not be animated because of your recipient’s email client version or because of non-default settings. So if you want to animate your logo, you have to ask yourself whether or not you are willing to risk the essential element of your design not showing up and what to do if it is not animated, after all.