i'm in south northants and my friend from my yard goes to norfolk on holcome (sp) beach, she tends to go for a weekend and finds a livery nearby to turn the pony out at
Well I'm in herts and we go to Holkam, Norfolk- it is the most amazing beach and definately worth the trip, takes us 3 hours each way but well worth it. The livery is called mill farm and indeed there are individual paddocks for turnout and stables for holiday horses and you can camp in the field. If you just day trip it cost £12 to park trailer/lorry at the beach car park.
Another vote for Holkham. We camped at a farm in Burnham Thorpe which was quite a long hack to the beach, but it worked quite well as the ride wore the horses out just enough so they didn't go mad!
I did quite a bit of research and there are very few other beaches that allow horses in the summer round this area. In Suffolk quite a few are shingle and the by-laws stipulate horses are only allowed from 1st October to 31st March, except for a nice sandy beach up near Lowestoft. Your other option is to head up to Lincolnshire, but not been myself so can't comment.