Another option is for the business to host the (open source) chatbox themselves, but the messages are stores and routed through a 3rd party. The chatbox is probably just a plug and play component that can be embedded in any page, and hosted statically by the business. Much easier than self hosting the entire messaging infrastructure.

This is one of the major benefits of having an open protocol like Matrix. The clients are separate from the servers. People with more resources and more expertise can host the servers, while regular users just need to download an open source client, and they can rest assured that the messages are secure.