Perry Webb has done a great job in pointing out the issue is the translator's choice of the many different targets for zeh. I'd like to finish off the answer by explaining what the KJV is saying and why I believe the option chosen by the NASB and KJV is the better one, and the option chosen by the ESV and NIV is worse. This is, of course, a matter of preference.
As many praises of God, there is a reiteration of the Exodus events. In this case:
O God, when thou wentest forth before thy people, When thou didst
march through the wilderness; Selah:
The earth shook, the heavens also dropped At the presence of God: even
Sinai itself was moved At the presence of God, the God of Israel.
The "earth shook" and "even Sinai itself was moved" is a reference to the earthquakes and cataclysms of the theophany:
And mount Sinai was altogether on a smoke, because the Lord descended
upon it in fire: and the smoke thereof ascended as the smoke of a
furnace, and the whole mount quaked greatly.
Therefore the idea of "Sinai moving before the presence of God" makes a lot of sense.
If we look at the cantillation, which are used to group words into logical units, we see the following graph for verse 68 (translation my own):
A: The earth shook
Ole We Yored A:
Tsinnor A: Also the heavens poured down
Tsinnor B: from the presence of God
Ole We Yored B:
Atnach A: This[zeh] Sinai
Revia A: from the presence of God
Revia B: the God of Israel
This suggests a parallelism were X quakes before the presence of God, and in this parallelism, both heavens and zeh Sinai appear as the ones in the first verse before the presence of God, with the second Atnach an elaboration (the presence of God, the God of Israel).
The alternative translations suggest that Sinai be grouped with "presence of God" in Ole We Yored A, which is not what the masorah shows. The Masorah shows that Sinai is also before the presence of God.