No, burning a media DVD is not supported by Vista's Media Player. You need a software like Nero, Roxio Media Creator, or Sonic My DVD to create a disk that is playable on a stand-alone DVD player.
If your computer came with the DVD burner, the chances are you already have the software on the computer. If you install the drive by yourself, the burner most certainly came with an installation disk for one of these programs.
If your Vista is Home Premium or Ultimate, you can also use Windows DVD Maker.
There's no copyright infringement involved if you created the footage itself.