Any home can have a radon problem, regardless of foundation type.
A basement provides a large surface area in contact with the soil material. Radon can enter through cracks in the concrete, floor-to-wall joints, control joints, etc. Exposure to radon can be especially high for people who use their basements as a living space.
Like basements, homes with slab-on-grade foundations have many openings that allow radon to enter.
Homes with crawl spaces (vented and sealed) can also have elevated radon levels. Radon-laden soil gases can be drawn into the home due to negative air pressure vacuums.