In order to install WordPress period you need full access to the CPanel on which the domain is hosted on. If you do not have full access to the CPanel there is no way to install WordPress on that domain.
The only way you could use FTP File Manager to install WordPress without access to the CPanel and Database is if you know the Username, Database, and Password used to setup the database. << That would be a long shot and is highly skeptical and may present a security risk.
Is it possible to install WordPress with ONLY a FTP login? I literally have no access to anything else except that..
Do you own the domain name? If you do own the domain name, you can request that your password be reset via going to http://www.YourDomain.com/cpanel (substitute "YourDomain" with the domain) and clicking on "Forgot Password". If you do not own the domain name and are setting up a WordPress site for Client, Friend, or Family member, I suggest you contact the person who does own the domain name and get the required information to access the CPanel and Database information.