Teams client would support shared computers if it installed to "Program Files", with updates via WSUS.
Teams is a program. It should install to "Program Files". Updates should be provided via WSUS.
Installing software to "application data" in order to sidestep administrator safeguards is a terrible idea, presumes no-one secures application data, wastes vast amounts of space, prevents use on heavily shared computers (i.e. education) and is an all round terrible idea.