“Can I ask for new carpet from my home inspection?”
You can, but that’s not what home inspections are for.
Home inspections are for you to find significant defects that you weren’t aware of when you made your offer.
Based on what you find, you can decide if you still want the home and if you want the seller to repair or credit you for those defects.
You obviously knew the condition of the carpet when you made your offer, right?
So it can be viewed as an act of bad faith if you made your offer aware of the carpet, presumably took that into account in what you offered, then asked for a discount due to something you knew in the first place.
My advice: take the carpet condition into account in your offer and don’t ask for any credit for it from your home inspection.