[RPG] Would using the Thaumaturgy cantrip give advantage on an Intimidation check

Yes, A DM could impose disadvantage and grant advantage on the same contest.

There is precedent for it. The Cloak of Elvenkind grants both advantage and disadvantage to a Wisdom (Perception) vs Dexterity (Stealth) contest. From the SRD:

While you wear this cloak with its hood up, Wisdom (Perception) checks made to see you have disadvantage, and you have advantage on Dexterity (Stealth) checks made to hide, as the cloak's color shifts to camouflage you. Pulling the hood up or down requires an action.

It is therefore possible to imagine other situations in which a Dungeonmaster could grant similar concurrent bonuses/penalties. Note that items which do this are very few. A ruling such as this based on a current situation should be just as rare. Advantage vs Disadvantage contests are pretty much a foregone conclusion, and almost beg the question "why bother to roll?"

No

The spell text reads:

You create an instantaneous sound that originates from a point of your choice within range, such as a rumble of thunder, the cry of a raven, or ominous whispers.

'Rumble of thunder' and 'the cry of a raven' are both non-verbal sounds. 'Ominous whispers' could be words, but the intention seems to be that they are not being used for communication (hence 'ominous').

Overall, the intention of the spell seems to be to create sound effects, not words.

This is supported by sage advice:

“Ominous whispers” : Can you form actual words using the Thaumaturgy cantrip?

I’d say no – context indicates to me it’s more for sound effects than communication.

An Alternative

There is in fact another cantrip that can perform this function, called message:

Range: 120 feet

You point your finger toward a creature within range and whisper a message. The target (and only the target) hears the message and can reply in a whisper that only you can hear.

You can cast this spell through solid objects if you are familiar with the target and know it is beyond the barrier. Magical silence, 1 foot of stone, 1 inch of common metal, a thin sheet of lead, or 3 feet of wood blocks the spell. The spell doesn’t have to follow a straight line and can travel freely around corners or through openings.

Therefore, allowing thaumaturgy to also do this would impinge somewhat on the usefulness of message.