[W]hy can't I access Apache from the intranet?
Because Apache is apparently bound to a network adapter that is not part of the intranet you are trying to access.
To be clear, 192.168.1.x and 192.168.56.x are separate networks in this case (assuming a standard class C subnet mask of 255.255.255.0).
The address Apache is using (192.168.56.1) is most likely a default IP for a virtual network adapter, used for virtual machines. Apache seems to have selected this adapter/IP in response to a generic Listen 80
directive (as you mentioned).
I remember seeing a similar address when I was playing with Vagrant a few days ago, could it be related?
This could be related. If Vagrant installed any virtual network adapters, unless you specifically removed those adapters, they are most likely are still active on your system.
If you are definitely not using that virtual network adapter, you may wish to remove it entirely.
In any case, you can bind Apache (as necessary) to a specified IP in httpd.conf with a more specific Listen directive e.g.
Listen 192.168.1.66:80
Please note, since you made comments about Vagrant, I am assuming your server/app isn't running in a VM. If it is (perhaps say you are running VirtualBox instead), you'd need to adjust settings for the VM to make it accessible to the intranet.