Try Whitefield Forest, Ryde.
This is a brand new site set amongst trees with very generous pitches and immaculate toilet facilities, and is very dog friendly.
It only opened in March this year and has already been awarded 4 pennants by the AA.
No shop on site, but Tesco is only 2 mins down the road and open 24 hours.
A 5 minute drive wil take you to St Helens Duver, the only dog friendly beach on the Island during the summer, and a large open area of national trust land with very little traffic so the dogs can run free.