i couldn't found any indication they allow use of composer, but you may need to fully deploy your assets for production locally, then upload the entire folder via ftp.
then if you have ssh access which is available from medium package and above, you will need to login to ssh via ssh client, ex. putty and run artisan commands to run migration.
and as i can see they use apache as web server so you gonna need to update the .htaccess with this
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteRule ^(.*)$ public/ [L]
</IfModule>
*note: there's $1 after public public/$1 but the markdown preview won't display it :D
this will redirect the requests to laravel public folder.