A friend of mine has the following problem. She is an accountant and her clients frequently give her their login credentials so she could download their accounting data from various financial institutions websites. Also she has a part-time employee who helps her with some routine tasks. My friend wants her employee to be able to download the clients' data without having to disclose her clients' login credentials.

Not unlike most of the commentators below I would also much prefer for everyone on the planet to have their own unique accounts for everything and never share. Unfortunately that is not the case. So please concentrate on the task at hand.

I am looking for a way to set up a local proxy server so it would automatically authorize the user's browser session at the remote web site with predefined credentials.

I need the credentials database to be stored on a server in the local network (not in the cloud and not on a user PC) so the solutions like various browser plugins would not work.
The configuration should support multiple accounts at the same web site.