Our webmail is a surprisingly popular service
Over in the lobste.rs reactions to my entry on how running your own email service is an artisanal choice, a number of people have been surprised that webmail is an important issue in this. To be honest, I'm surprised myself. We operate a pretty basic off the shelf webmail service, and entirely against my expectations it's enduringly popular with our users.
Our webmail's popularity is both in how often it's used and how many different people have used it. It's regularly a source of IMAP connections to our IMAP server and some people clearly make regular use of it. On top of the regular users, a significant portion of our user base has used it at least once this year, and quite a lot of people have used it (possibly infrequently) in several different months. A lot of people would notice and be unhappy if we stopped having webmail, some of them right away and some of them eventually, and probably a decent number of people would be quite happy if we improved it.
I do know that people use our webmail as an emergency fallback if their other clients aren't working. If your regular client or even device has fallen over, you can get on our webmail to do basic work with any old browser capable laptop or tablet or whatever, and that's most of them. I think that some people use webmail if they just want to quickly check their email without the entire work of firing up a desktop client, have it scan and filter their incoming email according to their classification rules, and so on. I suspect that some people find using webmail easier than setting up an IMAP client, especially if they're temporarily using what they consider a secondary device. Perhaps there are even people who prefer the webmail interface to the client interfaces they can get.
(One potential advantage of using webmail, especially on a mobile device, is that you don't have your departmental email 'on' the device, regularly being checked and perhaps nagging you with unread message indicators and so on. It's a lot easier to disconnect if you have to explicitly connect in the first place.)
I don't know if this is a common thing in places that (still) offer their own webmail, or if it's more specific to the department's population. A university department is certainly a different kind of environment than a company, for example, and our users can be highly varied.
(Slowly realizing how well used our webmail system is has been a bit of a learning experience for me. It certainly taught me that I didn't know as much as I thought I did about how people here use our services.)