I just upgraded from High Sierra to Mojave yesterday, and promptly encountered the Could not Check for Updates (I was on 10.14, I know there is a 10.14.2 waiting), and Could not Connect to App Store problem. It was strange as all my other applications, browsers, etc. were able to access and download stuff from the Internet properly.
I tried deleting the Security Trust Settings, and Disabling and Reenabling DigiCert CA Trust in Keychain Access, to no avail, even after rebooting. I was contemplating reinstalling from scratch when I read other posts that doing a restore won't help matters.
I even tried updating manually by downloading the 10.14.2 Combo Updater from Apple directly, but the problem persisted.
The mention of Firewall issues triggered an idea for me. I'm normally connected to the network via Ethernet. I don't have any proxies enabled in macOS, but I do have ClamXAV, Docker and VMWare Fusion (these create virtual bridged interfaces in the system) installed. As far as I know there are no firewall rules enabled that would cause problems.
On a whim I tried disconnecting the Ethernet cable and enabling WiFi, and lo and behold, the App Store and the Software Update Panels worked. After disabling Wifi and reconnecting via Ethernet, it seems to be working now.
Somehow the Software Updater and App Store uses a different mechanism to connect to the Internet, and it was stuck in some unknown state that persisted even with rebooting. Presumably switching the default interface for access to the network somehow reset the configuration.