Usually, if someone lives in a nursing home, they are showered a couple times a week, which would include a shampoo. If a patient is not able to walk or
stand, they are usually showered in a shower chair. The most popular shower chair is made of plastic, has wheels and handles (arms), with a seat that looks like a commode seat. This chair can be rolled into the shower easily. If a shower chair is not available, a shower seat can be used. This is a stationary seat that can be sat in a tub. With this option, the wheelchair bound person must be transferred on to the seat in the shower area, which can be challenging. These types of products are available in medical supply stores and on the internet, so you can pick the type of chair that is right for your patient.
If you want to wash someone's hair while they are in a wheelchair, this is much more difficult. If the wheelchair back reclines, then you can recline the person all the way down, with their neck and head hanging over a sink, and wash the hair that way. If they cannot recline, I cannot see how you could wash their hair in the chair without getting them (and you) soaking wet. If you absolutely must wash their hair in the wheelchair, you can try to buy dry shampoos, which are usually sprayed in the hair to absorb excess oil. If you really want to shampoo them, I would leave them in their bed, hang their head over the side of the bed (with a pan or bucket on the floor underneath their head), and pour water with a cup to wash them that way. Keep in mind that this is very messy and the bed can get wet, so be careful. You can put plastic underneath their head to try to protect the bed as much as possible. Hope this helps!