I saw a few other posts here on the forum about this, but my case is a simple situation in a fresh install. Should be easy to reproduce.
I just installed 12.0.1 on a RPI 4. Connected to my local network on the setup screen.
Tried to add an http source as some simple first testing but it kept saying it couldn't establish the connection.
Enabled logs and SSH, said it couldn't find the host. To make sure it wasn't just curl I pinged my local server manually ping the-nas.lan and that yielded ping: bad address 'the-nas.lan'. So then I knew it was system-wide.
I found that weird because in Settings>LibreELEC>Connections>mynetwork>edit>DNS servers, my 192.168.2.1 server was there, so it clearly got it from dhcp, just wasn't using it.
However in Settings>System Information>Network it presented Primary DNS: 8.8.8.8 and Secondary DNS: 8.8.4.4.
Ultimately I went and manually put my router's IP in /etc/resolv.conf (which previously only had the above two), and after that everything worked perfectly.
Standard behavior for the distros I've worked with has been to at least include the DHCP-provided DNS server in the ones used by the OS.
Must say though, as a first experience with LibreELEC I'm having a blast. Remote is working through CEC right out of the box, everything is good and snappy on the Pi4. Thank you for all the work you do so we can have this amazing experience.