How can I get an existing Vista installation to a "clean" state (without an installation disc)?

What you can do is somehow obtain some kind of Windows Vista installation media for your particular edition of Windows Vista. There are lots of places on the internet which have these files. You can do this, so long as you use the serial number given to you by Sony.

The actual serial number should be on the bottom of your laptop, including the edition of Windows Vista which you are legally entitled to have on your laptop.

I do not have a Windows installation disc (thanks, Sony!)

don't blame Sony, you went and bought this computer without making sure there are recovery and/or os installation disks included (which is a deal breaker for me!).

I tried creating a recovery disc by downloading and burning a DVD with the image here

this is not a recovery disc to restore your system, it's a bootable CD for the Vista Recovery Center (to be used for troubleshooting/maintenance).

if you want a recovery disc to restore your computer to factory default, you will have to obtain this disc from Sony tech support (usually at the cost of shipping).

alternatively, you can use your OEM license with any OEM Vista installation DVD matching the product pre-installed on your laptop (e.g. Vista Home Premium) and download the drivers from Sony's support page. this way you will have a 'clean' installation bar any OEM pre-installed bloat. when it comes to the product activation you may have to contact a Microsoft representative and explain the situation (usually very helpful folks), although there are ways to backup and restore the Vista activation files but i do not recommend doing this, MS don't like tampering with the product activation, you may end up with a 'stained' license and then you'll have to do a lot more explaining.

note: if you can't access the computer at all, you can use a Linux Live CD or BartPE CD to backup your personal data.