NASA expects humans to live on the Moon very soon.

[nzoomed]

I wonder how much each launch contributes to climate change?

Well the SLS rocket uses hydrogen as its propellant in the first and upper stage, so it only produces water.
The SRBs use largely aluminium and ammonium perchlorate with a synthetic rubber like material as a binder.
The SRBs likely produce some CO2, but its probably insignificant.