CC site at Beechwood Grange is within cycling distance, also if you wish you can leave your cycles at the Park & Ride at Monks Cross approx 5/10 minutes away. This is a large with site with approx 85% large hardstanding pitches. We stayed there for 10nights at the beginning of the month and although full most nights it never felt too crowded. Another smaller site which is nearer and within walking distance is Manor Farm (google york caravan sites for more information). This is an adults only site now but we stayed there about 5/6 years ago when it was a CL, but they have now expanded.
Hope this may be of help
William