Multiple Simultaneous Account Access Website

I need a program designed that will allow a user to connect to a website using multiple accounts. Currently I can log into the website in question using one account without issue. I have several user accounts for this website. I can only have one account logged in at any time. In order to use any of the other user accounts I have to log out of the first account and then log into the desired account. This is very inconvenient. What I want is some software developing that would appear very much like an internet browser window with five separate tabs. Each tab would have a user account on it and the ability to be consecutively logged in to the website. In order to do this, each tab must operate independently of each other using separate user agents and cookies. This will be a subscription service so individual user accounts will be created. I would like this to be web based so that the user can use the program/website when they are mobile. Please ask me any questions you may have. Thanks