If you’ve not read these threads then do, the web cam seems to be the most unreliable and poor quality part of the TR series..
http://sonytr.siliconpopculture.com/viewtopic.php?t=5163
http://sonytr.siliconpopculture.com/viewtopic.php?t=3529
Usually they just die but yours maybe be dieing, not a lot you can do but check connections and maybe see if it’s dieing is heat related (as it works for 5 mins).
You can get the covers off and to the camera without disasembling the screen, (see below topic), once off you could try with a desk fan or a/c unit blowing on it just to see if these cam deaths are heat related. if it works longer than 5 mins then it could be. Of course I’m assuming it’s working for 5 mins then not working again until you next turn on the laptop?
http://sonytr.siliconpopculture.com/viewtopic.php?t=5905