I'm specifically curious if they could produce fuel oil, diesel fuel, or aviation gasoline in the East Indies without having to export the crude oil somewhere else for refining.
Yes, there were oil refineries in the Dutch East Indies. Four of the Indonesian islands had major oil fields; Sumatra, Java, Borneo and Ceram (in the Moluccas). Of these, the first three had refineries.
Sumatra had the largest refineries, with several large ones around the city of Palembang and others further afield. Sumatran oil was largely produced at the Prabumulih oil field, and was transported to Palembang by pipeline. There were two major refineries at Palembang, one owned by Royal Dutch Shell and the other by Standard Oil. The Shell refinery was built at Pladjoe, to the south-east of the city, and had a maximum capacity of roughly 45,000 barrels of oil a day. It specialised in the production of av-gas. The Standard Oil refinery was built at Sungei Gerong, to the east of Pladjoe. It had a similar capacity to the Pladjoe refinery, but was smaller, with less duplication of works and machinery. There was a smaller refinery at Pangkalan Brandan, near Medan in the north of the island.
Borneo was split between British colonies on the north of the island and the Dutch to the south. Both of these had their own oil refineries. The British oil fields were at Miri and Seria (in Brunei), with a refinery at Lutong. The Dutch produced oil at Tarakan and Balikpapan on the island's east coast. Balikpapan was the only one with refineries, but these were of comparable size to those at Palembang. These processed some 5,240,000 barrels of crude anually. The Balikpapan refineries produced a wide variety of oil products, from fuel oil to light lubricating oils. Borneo crude oil could also be burned in ship boilers un-refined, though this did accelerate corrosion due to its high sulphur content. The oil from the Bula oil field on Ceram was generally shipped to Balikpapan for refining.
Java had several oil refineries, split into two groups. The first was the Rembang group, in the centre of the island, while the other was the Surabaya group in the east. The oil fields and refineries were mainly run by Shell. The Rembang group consisted of refineries at Kapoean and Cepu, while the refineries at Surabaya were in the suburb of Wonokromo. Java's refineries were the smallest, and had the lowest output.