Presidential authority to ban TikTok

What is the legal authority by which the president can order private companies to stop offering a particular product? All indications are that they will comply but what penalties could they face if they didn’t, by authority of what law?

The executive order to ban TikTok seems to have the force of law. I have never seen an executive order that does anything except dictate policy or execution of policy within the executive branch (not that I have studied them all). That seems to me the very definition of an executive order.

In the articles I have read on this, the focus is on the practical implications and reactions to it, but not the legal mechanisms for putting such a ban into place.

In an article in the Washington Post,

They will not prosecute but there is an implication here that they could if they wanted to. If so what law would they be breaking?

I was wondering this myself but since cooking with gas already asked,

Business Insider says he can, sort of: