I tried to get a Livery/Taxi fleet ID, and Ford rejected the idea of using it to order an F150 Lightning:
"Good morning, to purchase the F150 Lightning you do need a Commercial FIN, not Livery. The criteria to get a Commercial FIN can be found here - https://www.fleet.ford.com/get-started/eligibility-documentation/#commercial."
Commercial fleet needs you to have 15 vehicles already in service (proof required). OR order 5 new ones. But I heard the cap on ordering F150 Lightnings is 4. So it seems you cannot make a new fleet of just F150 Lightnings...