If you wanted to transport your own water, you could. However, because porters must carry water on climbs, hydration is significantly more crucial than hairwashing.
If you wish, you could also use dry shampoo. However, during a 7- to 10-day walk, it’s just one more thing to lug about.
If you’re willing to pay for it, you can!
If the daily morning bowl of water isn’t enough for you to wash your hair with, you’ll need more porters to arrange for more washing water.
Even portable shower rooms are available from certain trip organizers.
Is it also possible to buy dry shampoo? (powder to clean your hair without the use of water)