First of all you are not indicating a country, don't take this as face value and always check if it's applicable for your jurisdiction.
I don't have a formal contract with Company X, just agreements in writing on emails - does this count?
Yes, this would be roughly equivalent to a contract, and Company X would have had to abide to that.
the owner of Company X and sole website designer suddenly died (...)
It may depend on the type of Company it was, if you were contracting with the person, or with a company with identity by itself. I see a few options. If Company X itself survived (and was perhaps bought by Company Y), they should fulfill Company X obligations. It would also be possible that they are offering to provide the same support as Company X as a way to their clients. If Company X was itself not bought, you may have a case for requesting from Company X / the heirs of their owner the requested service / cost of the service non-provided (but how to measure which amount was for the initial design -obviously taking much more effort- and which for the maintenance?). I would consider the "free maintenance for 1 year" to be lost.
If we have to pay him and it is more viable to switch company rather than continue with Company Y, does he have any grounds to block the transfer?
No. It would have been preferable that there was a clear contract between Company X and you, but as I understand it, you would be the owner of the new website.
Please note there are probably two parts on this:
- The domain name
- The actual web page
The domain name should have been registered at your name (with Company X as the technical contact). That shows clearly that you own the domain name, and they have no grounds tob lock you from moving it elsewhere or pointing it to any other website (it's not always one "correctly", which then gives headaches when developer and website owner parts ways, but anyway IMHO it's very clear that domain atj786.com was intended to be owned by atj786, not Company X...).
Regarding the web page contents, that could be argued if Company X was to create the page for you (so you own the page, code, etc.) or just provide the service. Also, Company X could have programmed the website (or part of) for you, could have used a general CMS (like WordPress), an internal library it shares on all their websites. Worst case, you would only have the "design".
Still, this is a recently created site, and you can create it again with someone reputable, based on the work Company X did for you. And yes, I recommend you to avoid Company Y.