Does the business have to be based in Poland?
In theory no, however it would be bureaucratic overkill to register a UK company and then register a presence in Poland which you would need to do in order to get the pitch.
As for the pitch, where do I go to do that
In Warsaw it depends what sort of street it's on. Depends whether it's on a bus route or not. For one you go to the department of roads and highways and for the other to the city hall (easier, hence most of the people who sell veg on street stalls do it just round the corner from a bigger road). Outside big cities the rules may be different.
and can it be done with just basic Polish skills?
You would need a Polish speaker to help you through the process. My ex (who is Polish) did it once and it took a while. It's a little easier in one of those big markets (Wołumen, Rożyckiego, Braci Wagów if it's still there) where you just pay a flat fee to the organisers, however you would still need to be registered as a sole trader (plenty of information on the various steps to do that here in this forum).