What do I call a company that sells an item they don't make but under their name?

As well as Reseller/Rebranded etc., these products are called White-label products and the means of selling white-label service or simply white-labelling.

Wikipedia describes it so:

A white-label product or service is a product or service produced by one company (the producer) that other companies (the marketers) rebrand to make it appear as if they made it.

Investopedia's description:

A product manufactured by one company that is packaged and sold by other companies under varying brand names. The end product appears as though it is being made by the marketer, when in reality it is being created by the manufacturing company.

In this case, the manufacturer can concentrate on making the product or service and focus on cost savings, rather than worrying about marketing, which will be handled by the companies that will sell the product.

wikipedia: Rebadging is the term that is used for this kind of behaviour.

See also Camerapedia: Clones, Rebadges and Rebrands

There is some usage of the term rebadger : Google: is-a-rebadger

A shell company is a company that doesn't run a business as such. It exists only to facilitate legal and taxation arrangements.