You can get permits from the bike shop at Birches,just ask any of the staff & they’ll be happy to help
Not sure if serious, but you most certainly do not need any sort of permit to ride in Cannock Chase, with the one obvious exception of the one needed to park your car in the car parks!
Couple of notes from my experience:
The Northern loop (Follow the Dog) is less technical, nice and quick and lots of fun. It’s relatively flat too and you’ll find a fair number of families (including my own) battering around it. The last part down to Birches is a really good section and we’ve been known to ride this multiple times because it’s so much fun. My wife rode this and only complained to around a 5 or 6 on a scale of 10. Both kids loved it though.
The Southern loop (Monkey Trail) is distinctly hillier and more technical in places, with steeper climbs and descents and a handful of tougher technical elements that justify its red status. I did not take my wife around this as I value my marriage.
The two loops can be ridden independently or as a single large loop as required. There’s also a ton of forest road too, if you want to rack up the miles without taxing your brain.
All in all, it’s a great place and I love riding there. Have a great time!