It does not follow that without an objective foundation for morality that we do not have a morality; further, it does not follow that if we do not have an objective foundation for morality, and yet have a morality, that that morality is by default a subjective morality.
Theists and metaphysicians ask that we would have a non-human foundation for morality. They employ words like “objective” and “subjective” to forward their case. But we have no access to anything which is non-human and yet would give us sentences upon which to place our morality.
Perhaps it is true (though I affirm that it is not) that God gave his word to a privileged few; but we have no objective ground to judge if these speak true, even if they speak honestly.
But God is not the only means by which to argue an objective ground. Kant did this by positing an a-historical condition called human reason, and created an objective morality thereupon.
Yet I do not accept that there is an objective ground for reason, if by objective we mean not historically conditioned.
I demand a thoroughly naturalized morality, in keeping with Rorty, which entails we drop the words “objective” and “subjective.” These words belong to theology and modernity; yet their utility has waned.
There is no non-circular justification for morality.