Is there a modern browser for iOS 10.3.3 capable of supporting ES6+ Javascript? I have some iPads I'd like my staff to use. However, these are 4th generation iPads, so the latest iOS I'm able to install on them is v10.3.3. However, the website they need to access requires a more modern browser than the built-in Safari.

We tried downloading the Chrome app, but it seems the Chrome app is using the Safari engine.

Is there any way to use a modern browser (one that supports ES6+ Javascript) on iOS 10.3.3?
From what little I know about ECMAScript Safari on iOS (which all browsers must use for rendering web pages, as @Mark pointed out in his comment) some things are supported but many are not. And your guess is as good as anyone's as to why that is the case (not to mention off-topic here).

So you are likely out of luck for full ECMAScrip support on iOS unless you want to go the jailbreak route. We can't help with that and I don't know if there is a browser for jailbroken iOS devices that does support ECMAScript.