Why/How did Ted Faro have 'Omega' clearance?
Based on the rest of the team's surprise and according to the wikia, he did this without anyone's knowledge. How this happened is certainly a mystery.
Secretly acquiring a security clearance that was above even Sobeck's, he unilaterally deleted the entire archive.
The only relevant recording is one where Ted argues with Elisabet about the necessity of a backdoor in GAIA Prime. GAIA herself agrees, while Elisabet opposes it.
However, since the people who designed GAIA were unaware of the existence of Omega level clearance, it seems that it was unauthorized, or at the very least, kept secret from the rest of the team (and remained that way after Elisabet died).
Out of game answer
Per my question on HZD: Anniversary AMA, Ben McCaw confirms that Omega Clearance is the backdoor Ted mentions to GAIA and, rather than having GAIA create this, he would tasked his engineers to implement this without Sobeck's knowledge.
How did Ted Faro obtain Omega Clearance?
This is a question many players have thought about, and it is never explained in the story how Ted Faro obtained Omega Clearance. There is a datalog that mentions Ted wanting a back door into GAIA Prime, and fan theories believe Omega Clearance is that back door. but I am curious to how Omega Clearance truly came to be
Ben: Oh, that's a great question! Ted had had his engineers create the back door...GAIA wouldn't have done it for him!
....
Ben: Just to be clear, Ted would have had his engineers create the back door...GAIA wouldn't have done it for him!
(Partial) In game answer
As @Vemonus's answer clearly lays out, there is no in-game explanation of how Ted obtained Omega Clearance. The only solid evidence we have of Omega Clearance existing is the Emergency Record.
There is, however, another audio recording with Ted yelling at his engineers for not implementing a backdoor into the Faro Robots: Entangled Waveforms.
CODE EXPERT: The protocols use poly-phasic entangled waveforms. Quantum encryption, Black Quartz stuff, way beyond military-grade. That's what you demanded, so that's what we delivered.
TED FARO: You don't code something you can't crack. All we need is a back door. Upload the latest service pack update and the problem goes away.
CODE EXPERT: You specifically forbade us from leaving anything resembling a back door in code. "Every protocol to Black Quartz standard." Your words. Look. If you need me to fudge some projections, it's nothing we haven't done before.
TED FARO: I don't need fudged projections! I need a way to reassert control over the Hartz-Timor swarm!
CODE EXPERT: I don't know what to tell you, Ted. You're asking the impossible...
So naturally, Ted would have wanted a backdoor into an investment in the event he lost control. GAIA agrees to Ted's proposal for creating a back door, but Sobeck disagrees and we never heard anything else relating to this discussion.
Due to Sobeck's reluctant attitude towards this, Ted would have went around her and tasked his engineers to create Omega Clearance. Since GAIA initially agreed with Ted, I doubt she would have considered this a suspicious act