How can I access a 2012 Macbook Air storage when the display has failed?
You can boot the old Mac up into 'Target Disk Mode', by holding T at startup.
You will need a cable that connects the two together: this is a bit complex, because the 2012 MBP has FireWire and Thundberbolt 1 and the new Mac has no FireWire and Thunderbolt 3. So'll you'll either need a Thunderbolt 2 to 3 adaptor, or a Thunderbolt to FireWire adaptor.
When the Migration Assistant runs on the new Mac, you may be able to connect the other Mac via Ethernet (again with the relevant adaptor on the new Mac), to begin a transfer of the data. It's possible you might have to type in an admin password on the old Mac.
Removing the drive and connecting it via a USB-C to SATA cable is also possible.