One of the many detestable things about Chromebooks, which I personally wouldn't trust with an infinite number of the proverbial bargepoles, is that they expect you to do just about everything using cloud computing (or other web-based services) and, in particular, that they assume that you'll be using Google's own services. (i.e. most of the email apps available for Chromebooks expect you to be using Gmail, rather than any alternative provider).
So any
dedicated Chromebook app is likely to be based around Gmail. e.g.
https://itschromeos.com/2018/12/27/top-alternative-email-clients-to-gmail-for-chromebooks
Otherwise you're expected to use webmail, through your browser, to access the services of other providers.
However many (most?) Android apps will run under the Chrome operating system, so you could try something like Email by Edison (which generally gets very good reviews)
https://play.google.com/store/apps/details?id=com.easilydo.mail
or an Android app from your own email provider.
(e.g. I use GMX as my email provider, so I use the GMX app on my Android phone, which would probably work under the Chrome operating system too)