A clip of this can now be seen on YouTube. The vehicle was a specially adapted Toyota Hilux with some corrugated iron on top (to keep off the dropping fiery bits) and a water-cooling system for the tyres (to stop them burning, though they did). James May got amazingly close to the active crater and even managed to scoop up a bit of newly-spewed out lava.
I wonder though if this last bit was wise. Iceland Review Online reports that a British tourist, who stole a piece of lava from the volcano, has now sent it back to Iceland on the grounds that his life has been jinxed ever since. The University, who received the piece of lava, arranged for it to be flown out to the volcano and dropped back in, on the grounds that one does not trifle with the rocky powers that be, and referring specifically to the folklore associated with natural phenomena (see this longer Icelandic version of the story).
If you search the Iceland Review website, you will see that the whole Top Gear exploit was considered pretty controversial at the time, though clearly overshadowed by later ash-cloud related events!