can this be backed up and moved to new site

It can, but I don't recommend it. The Review List, Templates, and Badges all have export CSV file buttons on them, then you can import the CSV file into the new site. The problem is that if you want to auto-update new reviews then you have to setup the download anyway so it's best to just re-download them to being with. That way they for sure have all the same page names. So, I would re-download the reviews and maybe move the templates if I spent a lot of time on them.

Another option would be to copy the data out of the MySql database and import it to the new database. You would need to be comfortable with modifying your WordPress database however. These are the tables you would look for...

wp_wpfb_badges
wp_wpfb_floats
wp_wpfb_forms
wp_wpfb_getapps_forms
wp_wpfb_gettwitter_forms
wp_wpfb_nofitifcation_forms
wp_wpfb_post_templates
wp_wpfb_reviewfunnel
wp_wpfb_reviews
wp_wpfb_total_averages