What exactly was the function of a "Seaplane Tender" as used by the IJN and USN? By the size of the tenders themselves they don't look large enough to carry a very large compliment of seaplanes. We're they just for resupply and refueling? If not, how did they manage to carry a worthwhile number of planes?

Looking at the Navy's officially declassified record of naval aviation squadrons and air groups in WW2, we actually see a good listing of seaplane tenders and the aircraft they carried, by date.

The O2SU Kingfisher and the J2F Duck are some examples of such aircraft.

By war's end, the aircraft carried by seaplane tenders were primarily observation/patrol aircraft.

The advantage of a seaplane tender was that they didn't need to be very big. The ocean was the landing pad for seaplanes, and there was plenty of ocean to go around. Thus, seaplane tenders could launch aircraft on observation/patrol missions and recover them anywhere, without needing a big aircraft carrier or the creation and holding of an airstrip on an island.

How did seaplanes make it to the actual AO of a battle? We're they hauled, or did they fly out and refuel on site?

The low range of tactical aircraft in WW2 naval aircraft and the lack of aerial refueling meant that smaller aircraft were often ferried across the Pacific on such seaplane tenders, as well as on aircraft carriers/escort carriers.

Seaplane tenders could be put in logistical roles, carrying seaplanes out to the front.

What advantage would a seaplane actually have over a conventional aircraft? I'd imagine that they would save on limited carrier runway space, but did the planes themselves have any other advantages?

The problem with seaplanes was that they didn't have great aerodynamic performance. In order to land on water, compromises were made in their airframes, which meant they couldn't fulfill more dynamic flying roles, like mainline fighter aircraft.

The ability to land on water did have advantages. It meant that seaplanes could be used in air-rescue missions of downed service members. Also, since they were designed for long range patrol/surveillance in mind, they often had long loiter times which made them great for hunting submarines.

Of course, this also meant that seaplanes would eventually become obsolete. The introduction of the helicopter into naval aviation - and its versatility in air-sea rescue, anti-submarine warfare, and transport/cargo replaced the seaplane, and also eliminated the bulky equipment needed to launch and recover them (catapults and hoists).

Long range patrol was also replaced by advances in aviation after WW2. Far higher performance carrier based anti-submarine aircraft, like the S-2 Tracker, meant that seaplanes weren't necessary for immediate front line action. Extremely long-range flying patrol aircraft, like the P-3 Orion, meant that land-based anti-submarine/reconnaissance/patrol aircraft could patrol oceans all across the world.

As far as the Japanese Navy was concerned their development of the Chitose-class seaplane tender/carrier was a multi-role specialist support ship that could do several things, including deployment of midget submarines, retrieval, resupply, and deck launch of seaplanes (as opposed to the catapults that cruisers and up used), anti-submarine patrol (through the use of seaplanes to spot and bomb enemy submarines) and lastly the ability to convert it into an aircraft carrier if necessary. At the time of their construction, Japan was still paying lip service to the London Naval treaties, so much like the Mogami heavy cruiser design these ships were designed as a way to allow for quick conversion into a heavier military ship if and when Japan chose to abrogate the treaties.

It's important to note that most Japanese capital ships and cruisers had seaplanes equipped for use in reconnaissance, patrol, anti-submarine warfare, and search-and-rescue. However, especially on the lighter ships, capability for repair and rearmament was limited for these floatplanes. Seaplane tenders were able to provide replacements and replenishment near the front lines if a suitable base for aircraft or floatplanes had not been secured, as was the case in the Japanese conquest of the Dutch East Indies.