You should be able to install OJS without administrator access to the server. If your host won't let you SSH to the server, find an FTP client that supports recursive uploads (i.e. you can upload a directory and all of its contents, including subdirectories). Once this is done, the only tricky part may be setting file permissions according to the instructions in the README file -- you should also be able to do this via FTP. Everything else can be accomplished over the web.
If you're stuck at a particular stage of the installation, post some details here and I can provide some advice. (If you want to supply specific details like your hostname, please contact the team at firstname.lastname@example.org
rather than posting here.)
Open Journal Systems Team