Having worked for "managed services" firms in the past, I've looked for something like this but never found it. I haven't had the time or inclination to write such a thing since starting my own business, but there's definitely a market for it. It would definitely be handy for internal use inside an IT organization of more than 1 person, too.
I've seen too many kluged "solutions" using things like "Password Safe" that don't have strong (or any) auditing mechanisms. It's a huge pain to change all the passwords in the "safe" when someone leaves the company. Keeping the passwords stored server-side with granular access mechanisms and an audit trail would make life a lot easier in such an eventuality.
Features that I'd like include:
Authentication for individual users to the database such that an audit trail can be generated. Ideally the authentication system would use plain ol' HTTP authentication.
Your "access without signup" ("request" feature) sounds like using a unique URL as a "shortcut" to bypass authentication for a given credential that can access a single password. That sounds pretty straightforward to implement. When you create that one-time use credential you should have some kind of metadata to describe why the credential was created (for reporting).
Reports showing what passwords were accessed by which users to allow only the ncessary passwords to be changed when someone leaves the company. Once the data is in a database back-end this is easy.
Password expiration dates. I'd use these to drive scripts to perform automated password rotation and check-in of new passwords to the system. Often I've got things like service account passwords that I don't want to be subject to operating system password aging requirements but, at the same time, I'd like to have the passwords change once in awhile.
A database back-end with a web CRUD interface all wrapped up in SSL ought to work fine for this.
It shouldn't be too much work to knock together something quick and dirty, but making it really polished and clean (with a nice client API) would probably be some work.