It was never possible to like some site with Facebook "Like" plugin from email for security reasons.
But sometimes it is difficult to explain it to your boss, who wants something similar that others emails have. If this was your case, you could offer my answer to your management.
The following code doesn't make you a fan of the page but rather likes the site's home page, and it gives you a Like button and "X people like this. Be the first of your friends."
If this is what you want, please see the code:
replace http://eyeclarity.com.au - to your site
and if you do not like my like button, replace
http://www.4wdaction.com.au/_phpBB/download/file.php?id=68849
with your one
<a title="Like this on Facebook"
href="http://www.facebook.com/plugins/like.php?href=http://eyeclarity.com.au&
layout=standard&
show_faces=false&
width=450&
action=like&
colorscheme=light&"
>
<img title="Facebook Like Button"
src="https://upload.wikimedia.org/wikipedia/commons/1/13/Facebook_like_thumb.png"
alt=""
height="50"
width="50"
/>
</a>
jsfiddle
aforementioned code will redirect you to a link, that looks like this
https://www.facebook.com/plugins/like.php?href=http://eyeclarity.com.au&%20%20%20layout=standard&%20%20%20show_faces=false&%20%20%20width=450&%20%20%20action=like&%20%20%20colorscheme=light&
Now, you can try clicking it and see some animation that Facebook is offering.