Update: My new favorite plugin is Duplicator which makes quick work of gathering a full site backup (files and database). I have only run up against a few hosts that were not compatible with the plugin, it is super easy to use and is now my go-to for making full manual backups. A video has been posted which shows how to take a manual backup using the Duplicator plugin
So a while back I was faced with a situation whereby I was working with a new client. They had decided stop working with their current Web services provider for a number of reasons. However the provider was not being very helpful in the transition. They repeatedly failed to provide accurate access information or a file and database dump. The client asked if there was anyway we could simply take the Website rather than waiting for them to give us the needed access. Upon doing a quick Google search I found that I was not the only one looking to transfer a WordPress Website without FTP or MySQL access. Since that time we have had to do it twice and did so with a simple backup plugin. Administrator access is required, but of course if it wasn’t we would all be in real trouble.
The catch here is that the plugin requires certain server settings and specs to be met and if you don’t have FTP access you certainly do not have SSH access. So there is some luck needed in this game, as certain folders will have to have particular permissions and certain toolsets will need to be installed on the server. However I have found that most WordPress hosts meet the minimum requirements and the only time this didn’t work was when the site was on a Windows box. There are certainly a number of backup solutions out there, the following is just a favorite of mine.
This is the go-to because it is free and a great backup-tool. You can easily backup both files and database to a local directory for download, to Amazon S3, Rackspace Cloud, DropBox and many other storage solutions. The result will be a standard dump of all site files and a sql dump which can be used to reinstate the site on a new server. Most typical WordPress hosting environments will support this plugin without issue.
We also use this tool to run regular backups on some WordPress sites and I have written on the importance of maintaining and verifying a regular Website backup routine.