As a follow-up to this, we have confirmed in the community that this is the case.
If you install https://play.google.com/store/apps/details?id=com.google.android.webview then the app will start working.
Why this makes no sense is because installing the above does nothing. On GrapheneOS, you can only use the pre-installed Vanadium WebView - you can never use another WebView implementation, no matter what you install.
The Proton Mail app is just checking for the existence of com.google.android.webview
and only allows you to preview e-mails when it detects it as installed, even though it hasn't functionally changed anything in the slightest. It's a hardcoded requirement that makes no sense.
My hope is that at the very least this will only apply to the beta version of the app and not the full release.
Marking as solved as this is clearly an intentional decision by Proton Mail which can be trivially bypassed. There's nothing for GrapheneOS to do here.