A word of warning - don't dump neat veggy oil into a PD engine.
However it will probably run OK up to certain dilutions in older style TDI engines.
Remember that biodiesel and vegetable oil are two completely different things, you'd need to use biodiesel in PD engines. The handbook that came with my 03
Ibiza TDI Sport (PD) said that Biodiesel was OK to use so long as it met the relevant EN specs. However I believe in subsequent years they removed this advice from the handbooks of PD engined cars.
The other alternative is to buy a kit that gives you a second tank in the car which holds veggy oil. Once the engine has warmed up on normal diesel, this is passed through a heater of some sort so that the veggy oil is at a more useable viscosity; it then needs to be purged through with normal diesel before the engine is switched off.
I believe this has been used OK on some VAG PD TDIs, I was looking into this last year, but I'd be happier waiting to see more long-term reports on PDs. Plenty of kits in use on other cars though.
Useful info:
The "Goat Industries" site:
http://www.vegetableoildiesel.co.uk/
..and their forum:
http://www.vegetableoildiesel.co.uk/forum/index.php
Journey To Forever site - loads of info:
http://journeytoforever.org/biodiesel.html
There's lots of other sites around too, just google for "biodiesel", and "vegetable oil diesel".