Email doesn't get opened in a browser per se (only in case of a webmail solution, or if the email clients fires up a browser).
So, you really should ask if all email clients will handle a gif file in a html encoded email message.
The answer is a very loud NO.
There are email clients that don't even decode html at all.
That said, most users aren't using those text only email clients anymore. But even those using email clients that parse html, the diversity of clients is significantly broader than with browsers, so don't expect anything near "perfect" everywhere.
Keep it simple! The fancier you try it to make, the more likely it's to blow up somewhere.
Some services that allow you to send out email to large lists have preview tools that allow you to see how commonly configured email clients will show your message. (There are probably others, I've only ever used mailchimp myself)