If you have your own mail server and view your mail locally on a client like thunderbird, you can’t really view your mail on any machine in which you don’t want to be installing an email client on or one that isn’t yours. Are there any self-hosted webmail options in which you can log into your email on any machine without using a 3rd party service?
Other people already mentioned Roundcube and Snappymail, which are good options already.
There’s also Cypht if you want a different approach. It combines multiple accounts into same interface so you can have a unified inbox.