I think I might finally have 3G working now. I have been having email conversations with Vodafone although they have been suggesting things that I've already tried, and it's impossible to register for an account on their site. Happily, I did some sniffing and found some good tips.
First of all, I connected via wi-fi and made sure that the firmware of the Vita was up-to-date, which everyone should do automatically anyway. Once updated I switched off wi-fi, went to the Network Operator thing and changed the APN to Vodafone. After that I went to the Vodafone site on the PC and instead of trying to register for an account (currently borked) I just went here... http://www.vodafone....nline/index.htm
...and chose "Top up online" before filling in the top-up form as usual. If this works, you'll get a message through to your Vita. It helps to have wi-fi switched off during all of this just to make sure that you're definitely receiving the confirmation message via 3G.
Long story short, registration on Vodafone is shafted and it's easier just to do the quick top-up. After doing all of the above I was able to use the browser whilst wi-fi was still switched off, so it's all good. Oh, and of course you should browse the Vodafone site on a PC rather than the Vita.
EDIT: all that said, I did ask for the SIM to be activated during my email tennis with Vodafone, so I'm not sure if that's part of it too...