Well I'm guessing No, no PPE (ASP, PAVA/CS, Cuffs/Stabbie) - Why would you need it? . I'm also guessing, No radios [excluding at events], and no, they don't patrol, they aren't PCSOs / PCs.
I was a former SYP Cadet back in 2005 to 2007...
SYP Cadets back then had access to pool sets of body armour we also received SOME Airwave training and utilised the Pool ''Special'' airwave terminals, now things may well have changed by now.
The SYP Cadet scheme is one of, if not the best in England when it comes to the opportunities and experience it offers to young cadets, I wish you the best of luck as the two years I spent with them really did help me into eventually becoming a regular PC.