Hi Charles,
Thank you for posting to the community. I would be happy to help you regarding your issue.
Note: As of 0.1.0-beta.3, Flarum utilizes Composer to manage its dependencies and extensions. This means that Flarum cannot be installed on hosts without SSH (command-line) access. Rest assured that we are planning to address this in the future and make sure that Flarum is accessible to everyone. In the meantime, if you want to run Flarum, you will need to find a host that permits SSH access.
According to Flarum's website, after Beta 3, they started using composer, the php package management software. Since then, you can't use or install Flarum without shell access.
There may be another way of doing it (if someone did find a solution to it) but official ways don't give any ways of installing Flarum without shell access.