Replacement of the 2zz-ge waterpump is a straightforward but fiddly job to do. This specific pump has a cast impeller suited to function properly in the high revving 2zz-ge application, and they hardly ever fail or leak. In case you suspect a leakage, it may well be just the seal between the pump and the timing cover. The latter is grooved and provided with a rubber o-ring or bead gasket. Either way, the pump needs to be removed to sort a leak.
The pump is (still) readily available at Toyota dealers, and has part number 16100-29146. Make sure it hasn’t a sheet metal impeller as that is not for the 2zz-ge. The pump may set you back 120-200 eur/dollar/gbp, cheapest in the US, expensive in europe. The cast AISIN pump housing has changed a bit over the years but is still interchangeable.
Remove the rear undertray and put the car on jackstands or lift. Removal of the right rear wheelbay gives some more access.
Consider breaking the pump pulley bolts at this point, as the serpentine belt locks the pulley for you. Note the belt is still under tension so just break the bolts free and do them up again.
From below, use a large spanner or torque wrench with 18 or 19 mm low profile socket to rotate the belt tensioner clockwise and remove the serpentine belt. (Ref: How to replace a Denso alternator in a 2004 Elise (2zz-ge)
If the bolts are not released yet, find a suitable pin to lock the pump pulley by sliding it in a hole in the pump casting to be found at 12-o-clock top position. Undo the the pulley bolts and remove the pulley.
In case the front clam happens to be taken off, drain the coolant at the sill pipe to radiator hose connection below the crash structure. Otherwise prepare a bucket to catch coolant seeping out of the pump gasket. Make sure the crankshaft pulley and A/C compressor (if present) is wrapped in plastic to protect it for a flood of coolant.
Undo some pump bolts until coolant is released and wait for some 4 or 4,5 litres of coolant to drain in a controlled fashion. Then remove the pump.
Clean the gasket groove and replace seal and/or pump, and re-install in reversed order.
At this point you may consider to mount underdrive pulleys on either the alternator and waterpump, adding 50mm to your serpentine belt, reducing the revs of the alternator and pump on track (BUT in traffic jams as well which is a drawback).
Refill with red OAT coolant, i prefer the use of a venturi vacuum filler. Test drive while inspecting the coolant level, in particular after the thermostat has opened above 75 or 80 degree C.