p.p.s. Further update thanks to @fribygda
You can search for custom emoji too.
When you select a custom emoji its alt text surrounded by colons will appear. To search for it, you may have to type in the custom emoji's alt text without the colons :: around it.
For example, searching for blobsad will show posts and profiles with :blobsad: in them.
(Some apps may allow searches with the alt text's colons included, but the web interface doesn't, which is why I suggest removing them.)