Suffolk has 11 recycling centres, with a compulsory booking system in place. Although you're required to include your address on the booking form, I can see nothing on the relevant website to suggest that waste from people living outside the county won't be accepted.
The neighbouring county of Norfolk doesn't use a booking system, so the staff at recycling centres in the county won't know where a vehicle has come from anyway.
Essex operates a booking system, primarily for residents of the county only. However their website states:
"Residents from nearby areas in Suffolk or Hertfordshire can also visit the sites. This is thanks to reciprocal partnerships with Suffolk County Council and Hertfordshire County Council allowing cross-border use"
Cambridgeshire's website states that their recycling centres are intended for use solely by residents of that county but, as they don't operate a booking system, there's no way that their staff could tell if someone had driven there from further afield.
So it seems that, in effect (either because of reciprocal agreements or a lack of any method of enforcing the rules), all residents of Suffolk are free to use their nearest recycling centre, irrespective of whether that centre is actually located within the county.