So the 'webapps' folder is their webservers' document root?
If so, follow the instructions in the docs for "installation inside the document root".
The idea of having a "public" folder is to make sure all your code is outside the webservers document root, so that if it is compromised, no code that is injected can be called from the internet.
If you can't make that separation, because all you have is a docroot, then having a "public" folder becomes pointless.
But, since you say you have a folder called 'webapps', can you create other folders at the same level? Can you have /fuelphp and /webapps/mystartapp? If so, you can still make the split.