Originally Posted by
Jhieminga
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.
I may be wrong but I don't think the situation the OP is describing is the typical hosting provider relationship that you describe here. It sounds to me as though it is a small local company that has been paid to design and maintain the website, if so the company may have registered the domain name themselves in which case they may well own the rights to it (dependant on the terms and conditions of any contract they may have with the OP). As you say it seems unlikely that they should have any reason not to agree to transfer it, unless of course it is a particularly unique or desirable domain name in which case it could be a different story!