We know of the following hosting companies that are compatible with Domain Mapping System.
If your hosting platform offers support for Domain Aliases or to control the Domain Root for each domain handled by your server, then you should be able to setup Domain Mapping System properly.
Anybody can setup Domain Mapping System with the right server access if you are a system admin or have a VPS, Dedicated Server, etc.
We have no affiliation with any of the companies listed. We have received requests from many users of Domain Mapping System over the years, and these are the companies we've verified are either fully supported or not.
VULTR (w/ CloudPanel)
Any hosting company providing cPanel access with an Aliases option.
Any VPS or Dedicated Server with root access.
Any managed host that does not allow mapping of multiple domains to a single WordPress installation.