No, you should never go barefoot on an inflatable bouncy castle. Operators and safety standards universally require socks (minimum) or soft-soled shoes. Here's why:
Vinyl/PVC surfaces absorb solar radiation. On a sunny day, surface temps can exceed 50–60°C (122–140°F)—hot enough to cause second-degree burns on bare soles in seconds, especially for children.
Water (from drinks, sweat, or intentional "water slide" use) makes the surface extremely slippery. Bare feet have poor traction on wet vinyl → falls, collisions, head/limb injuries.
Any of these can puncture or lacerate bare feet.
Verrucae (plantar warts), fungal infections (athlete's foot), and bacterial transmission spread rapidly on warm, moist surfaces with barefoot traffic.
Most commercial rental contracts and insurance policies mandate footwear rules. Violating them voids coverage and endangers users.