Apple - How to enable search when using Emoji keyboard on iPhone?
You can't use search when looking for emoji in the stock iOS keyboard.
iOS uses predictive text to determine what emoji to offer you based on the words you're typing. While this isn't what you're asking for, you can see this iPhoneLife article for more info.
To search by emoji, you need to install and use a 3rd-party keyboard. Many developers offer keyboards that incorporate emoji search. I use SwiftKey (not associated with the software or developer) for typing on all my iOS devices.
Apple has indicated that the upcoming iOS 13 will offer:
Word and emoji suggestions
If you need to correct a word, simply ask and you’ll be presented with a list of suggested replacements.
Another alternative is Searchmoji.
It's a free keyboard that allow to search emojis with keyword. The best part is that Searchmoji doesn't require full-access to work (like all the other third party keyboard out there), therefore there's no way you data will leave your iPhone.
I am a developer of this app.
More info here: Emoji keyboard app