Moving WordPress from host without FTP?

I have a website hosted with service A which do not provide FTP access, and I would like to move it to another host B which provides FTP access. Is there any way to move?

Traditional Archive,Download, and Upload is useless, since site is pretty large(2 GB).

I’ve tried getting some most-popular plugins, but none seems to do the job. Also max_execution_time is 30 secs

Any suggestions?

in your case – i will suggest you to use this plugin :

http://codecanyon.net/item/super-backup-clone-migrate-for-wordpress/12943030

This plugin can make the backup of the site on Google drive,Dropbox amazon and etc, and after that you can sync your site with cloud (your old site and new site will be connected via cloud). So you don’t need to download and upload the site again.