Are any of the vegetable cooking oils a problem for celiac disease (gluten-sensitive) people?

No. There is no gluten in cooking oil. Gluten (from latin gluten, "glue") is a protein composite found in foods processed from wheat and related grain species, including barley and rye.
Depends... Pure vegetable oils are ok. But look out for the term hydrogenated oil on the label. Hydrogenated oils mean that they are mixed with something, usually soy or gluten based. If you see that on a food that is labeled gluten free, it must be mixed with soy. If the food is not labeled gluten free, it is best to check with the company.