Yes, you can send a special request with the booking, but you should be honest with your customer that it is a request and not a guarantee. When you make the booking, there is a notes field where you can type things like a late arrival, a quiet room, twin beds instead of a double, a high floor or a honeymoon stay. That text travels with the booking to the hotel supplier and on to the property itself.
What happens after that is out of our hands and out of the supplier's hands too. The hotel decides at check in, based on what rooms they have free that day. Some suppliers also limit how long the note can be, or will only pass it on in English, and a few older connections do not carry free text at all. Where that is the case, you will see the field is not available for that particular hotel or room, so it is worth checking at the time of booking rather than assuming it went through.
A few things are better handled as part of the booking itself instead of as a note. Bedding type is often sold as a separate room option, so if twin beds really matter, search for the twin room and book that rather than asking for it afterwards. The same goes for extra beds, cots and meal plans, which usually have their own rates. Anything that costs money, like early check in or parking, is normally settled directly at the property.
For a stay where the request is important to your customer, book the correct room type where one exists, add the note anyway, and then ring the hotel a day or two before arrival to confirm it was received.