It depends on the faction type.
For example new mining factions get the DC for groundhogs and FK-7 prospecting vehicles right from the start.
But: Factions that actally *can* produce won't be given any DC from the start at all. Make sure you have a DC-rental arranged from one of the DC holders before even considering to found a new production-type faction.
Governments can't be formed as a new faction, so no, they do not receive any new datacards upon conversion. They receive the ability to construct government-only facilities and the like, but nothing that involves datacards.