Is there an application for helping my clients to be aware of mails that arrive please? If there is no application like gmail or outlook, is there another way to alert about the arrival of a message in the mailbox without having to go there to see?
If I understand you correctly, you want a notification for a new mail but without an external program? Well, unfortunately, that is not possible.
My advice would be: Each phone has a mail client nowadays - they should use that and then log in to the webmailer. They could also install a mail client on a computer. In my experience, that really makes the work much easier. And yes, it does not have to be Outlook (which I have not used for 20 years due to reasons )