It is not possible to install Paratext without administrator rights. This is a requirement from the Windows OS that you can’t install programs to locations accessible by everyone on the machine without admin rights. This is very unlikely to change.
EDIT: For clarification, I should mention that installing and updating Paratext will need admin privileges.
Also note that Windows XP will not be supported starting from Paratext 8.0 as Microsoft stopped supporting it in 2014.