You're definitely most likely going to have to shoehorn the Play Store onto there.
"Run a current browser" is kind of a stretch - it says it's running Android 4.2 which is coming up on 3 years old. That gets you the buggy system WebView and not Chromium. You could install Chrome if you could get Play Store access onto it, but that won't help you if you're installing an app that is just a WebView wrapper.
They replaced the WebView with a (fall 2013-ish release of) Chromium in 4.4, and in 5.0+ the WebView component is kept in line with Chrome automatically by silent Play Store updates.
I also would _never_ expect any sort of OS update for any device like that, but I assume if you're looking at specialized hardware like this, you're already used to that.