Typically a hosting provider will register a domain on your behalf, which means that you'll have the option to take that domain name with you when you move to a different provider. Post-GPDR the WHOIS lookups don't show the full registration data anymore, so you can't check that unfortunately (my domain name has been registered by a local company, but my name and details used to be visible in there as well, as I am the one paying for it). If they did this the correct way, the domain name is technically yours.
If and when your IT literate person has identified a new hosting company, they should be able to look into the details of transferring the domain name to point to the new host. I would ask them to look into this. As long as your still on speaking terms with the local company I wouldn't expect any problems. As mentioned, it would only cost them money to hang on to a domain name, so why would they want to hang on to it?