Probably your best bet is simply copying all app files, but be aware it may take quite long (many files) if your only access is FTP, with risk of incomplete transfer. May be better (but not necessary) to transfer a single compressed archive file and extract it via PHP zip extension or exec() and tar command if available (you can find many tutorials on the web). Last but not least, you could try to run composer via PHP script - take a look here for example - but that could be much harder than expected (it didn't work for me some time ago because the hosting service had proc_open disabled).

Also, in your case you most likely have permission to access only your own web root directory and you can't change the document root configuration, therefore probably you won't be able to place "non-public" elements outside the document root as recommended, so at least remember to set file permissions properly.