Prayer chairs are on trolleys in the walk ways inside the Haram. If you don't see any you could ask the workers in the area and they will gladly oblige. A small tip would be in order.
Always check the chair before you sit as sometimes the are damaged and one could get hurt.
Another option would be to buy one - it cost 25 Riyals and one could take it around to use when needed. At the end of the trip you can leave it in the Haram for others to use.
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