Spain has an extensive network of campsites, from large coastal resorts along the Mediterranean and Atlantic coasts to smaller inland and mountain sites geared toward hiking. Most accept dogs as a matter of course, though — as with hotels and rural houses — policy, fees and any breed restrictions are set by each site individually rather than by one national camping body.
The near-universal practical rule across sites is that dogs stay on a lead within the site itself, for the same reasons it applies anywhere: shared pitches, children, and other guests' pets in close proximity. Many sites also cap the number of dogs per pitch and expect waste cleaned up immediately. Some of the largest coastal resort-style sites restrict dogs from pool areas and the busiest family zones, which is worth checking specifically if that matters to your stay.
Inland and mountain sites, away from the biggest coastal resorts, tend to be the most relaxed about dogs and offer the most direct access to walking trails straight from the pitch — worth considering if the trip is built around hiking rather than a beach holiday, where peak-season restrictions are more common.