This may be a good time to speak with the vendor. It is perfectly possible to create an MSI that can install "per user" to the user profile without the user having admin rights, but if you are unable to install the MSI as a non-admin user then something else must be going on that does require admin rights. They should be aware that deployment systems target computers and not specific users, so if their MSI is not able to handle that scenario, they are not doing a professional job of packaging their add-on.
You originally suggested using SVS as a capture tool to identify the files and registry entries, or you could use a free capture tool such as InstallWatch. Once you know what is actually being installed at the file and registry level, you should be in a better position to create a solution. There may prove to be some "machine" components such as HKLM registry keys or files placed under Program Files or in the Office folders instead of being located in the user profile, which requires admin access.
Ultimately, you may find the exact registry entries needed in HKCU to complete the install of your MSI, so that you can deploy to a machine, then add an active setup component to install the "per user" content at logon.