Unfortunately you don't tell us which HP printer it is, so it's not possible to give you an answer.
What Model Number is it?
So....Best bet is to go to the HP site,
Look under the support tab for YOUR printer, to see if HP have issued an updated driver for Windows 7. Without that, I suspect that you are off for a new printer. You may find that the driver from Vista works (if there is one)....but no guarantees.
Like Vista, I would have thought that if there was a driver Win 7 would have installed it automatically for you when you plugged the printer in.
HP do provide extensive help, including updated drivers for many of their printers, but unfortunately printers come and go so quickly that often (usually) in my own experience, it's cheaper and certainly FAR less hassle to just go and get another printer.