When you are stuck with your vagrant machine the way described above there is no need to boot in gui mode (and is impossible without an X server).
While your VM is booting, in a separate terminal window, just find out the id of the running machine.
vboxmanage list runningvms
This will result in something like this:
"projects_1234567890" {5cxxxx-cxxx-4xxx-8xxx-5xxxxxxxxxx}
Quite often, the VM is simply waiting for you to select an option in the bootloader. You can send the appropriate keycode (in the case, Enter) to the vm with controlvm:
vboxmanage controlvm projects_1234567890 keyboardputscancode 1c
That's it. Your virtual machine will continue the boot process.