I've had this exact same problem with my Sienna (2002 also). I can get my van to run by keeping my foot on the accelerator until the car warms up, (this seems to work faster if you can actually drive for 30 seconds or so). Check your idle control valve as I can also get the van to keep running (a bit shaky though) if I disconnect the tubing attached to the idle control valve. I'm not sure yet if replacing this valve will fix the problem as we just figured this out this past weekend.
I've had this problem intermittently for almost 2 years. Toyota keeps telling me it's my O2 sensor and has replaced it a few times. It's not! Good luck.
Monday, May 18th, 2009 AT 9:08 AM