Prayer chairs are on trolleys in the walk ways inside the Masjid. 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 you could get hurt.
Another option which would be cheaper is to buy one - it cost 25 Riyals and you 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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