I'm not sure whether to take a sick day or use my vacation days I have left.
Company policy states no vacations after two-week notice
Well straight away let's rule that out: you cannot use your vacation days (of course, you could still ask, but that's unlikely to be a possibility).
I don't think a company can really forbid you from using your sick days, and people do get sick in their last two weeks, just as they get sick on their first day of the job, as unfortunate as it is. You could definitely use a sick day. It will look bad, but it won't be the end of the world and I don't imagine it would be held against you for longer than your remaining time at the company. Sure, everyone will gossip on the day, but that's probably as bad as it gets.
I was made an offer and a start date. They want me to come in and do paperwork, meet colleagues, managers, etc..(lasting two hours)
As Rachael states in their answer, this kind of admin stuff should really be done on the first day. I can't imagine any business that would expect you to be productive on day one.
You should contact your new employer in the same manner you have been (e-mail, phone, whatever) and simply explain that the start date you agreed to was the absolute earliest you could start, and you haven't taken a vacation week (something people sometimes do) between jobs. I'd imagine once you contact them, they should be flexible in allowing you to do this on the first day, and I don't see this being held against you.
Ultimately, would you rather (potentially) upset your new employer by stating you won't come in till your start date, or (potentially) upset your old employer with a sick day? I don't think either situation will be the worse to find yourself in, and likely after a week it'll be water under the bridge.