Our cat is fully vaccinated and has a passport, but the old irresponsible owners were meant to have sent it me (we used to be friends) but they haven't, and they are ignoring any messages I send them asking for it. How do I go about getting her a new one, as I have no proof of vaccinations etc?

You would have to start from scratch, unless the previous owner gives permission for your vet to contact their vet to get the history, including details of the vaccines given, and the blood test results for the passport.
If the cat has a passport it must be chipped so you have to get that transferred, you probably need the previous owner to sign a paper for that transfer but if you phone the chip company with the details they could maybe help you more there. Your vet can scan the cat to give you the chip number which you will need to do a transfer.
It is not impossible that no information has been forthcoming because the cat didn't atually have a passport?
Or because your former friend is just being unhelpful.

I know she's microchipped because the vet checked for me last week, but the old owner said they would send me the passport yet hasn't bothered, I've left her messages on facebook etc, which I know she is getting because shes been writing messages on there etc. I can't get the old ownder to sign anything because she no longer lives near me (She moved back to UK)

Have you tried asking your vet to get the cat's previous vet to fax the cat's notes to him? Then you will have a full record of her vaccinations etc. My new vet did this with my cats when I moved to a new area - I was not asked to give permission, I had no input in the transaction, it was vet-to-vet. Might be a way of bypassing the recalcitrant previous owner (assuming you know who the previous vet was).

Yes but in that case there was no client confidentiality issue with any other owner.