Outgassing can be a concern.
Natural rugs may be your only option, but I'm sure they are not cheap.
Look at www.gaiam.com for the types of carpets/rugs that they carry and try to find something similar locally.
Your time frame really limits your ability to research and find someone to do the work, properly.
How are the floors underneath? Could you luck out and find hardwood under the carpeting?
Or, maybe look into having hardwood or one of the laminates done instead of carpeting (easier to keep clean too)
I'll see if my boss has a source for natural fiber carpeting.