Let's assume that there is a lake that has formed from snowmelt and rainfall. The lake has never been in contact with other bodies of water. How might this lake become a complete ecosystem?
Fish lay eggs. The eggs sometimes adhere to the feathers of water birds. The birds fly from one body of water to another, and some of the eggs survive the trip.
This occurs with farm ponds -- despite never being stocked, they wind up with populations of (in the American south, anyway) bream and bass, as well as inevitably snapping turtles that feed on them (the turtles travel cross country on their own feet, in many cases).
Flying insects, of course, fly -- and many of them breed in water (mosquitos, caddisfly, just to name a couple). They provide food for the fish, likely before the fish are introduced.
Amphibians arrive the same way fish do (as egg hitchhikers), or travel overland. Water plants also travel by hitchhiking -- either seeds in the gut of a duck or goose, or fragments stuck to feather or legs that manage to survive and root.
I'm guessing that your question about endorheic basins is about the buildup of salt and other minerals. Certainly there are many aquatic animals who live in fresh water and many that live in salt water, but not many who are okay with a change from the former to the latter over time.
The solution to that (no pun intended) is a lake that doesn't get the mineral build up of most endorheic basins. For example:
Crater Lake in Oregon, USA, also mentioned by Michael Richardson. Despite the fresh water coming in, "the lake has relatively high levels of dissolved salts, total alkalinity, and conductivity. The average pH has generally ranged between 7 and 8." Even so, it supports aquatic life, including fish.
The largest lake in the world, the Caspian Sea, between Europe and Asia, is also an endorheic basin. "It has a salinity of approximately 1.2% (12 g/l), about a third of the salinity of most seawater."
Modern times, including pollution, have harmed its ecosystem, but it flourished for quite some time. Its fauna include: turtles, mussels, gulls, terns, and seals. Even other aquatic mammals.
Technically it doesn't meet your requirements for never having been attached to other bodies of water. But it has been landlocked for 5.5 million years, which should count for something. The fact that it supported a wide variety of plants and animals a century ago is more about its current ecology than about its past.
Your question may also reference the "seeding" of the lake with various fauna and flora. Zeiss Ikon talked about how to get fish eggs (without human intervention) over land. But I'll also maintain that your requirement of "a complete ecosystem" doesn't mean it has to have fish. Some aquatic fauna for sure, but there are many many choices.
Plants can adapt, and their seeds or even pieces of root can be carried by animals that travel by land or water. In addition to fish eggs, traveling animals can carry snails, small reptiles or amphibians, or their eggs. Insects can fly or crawl or hitch a ride.
Given enough time, an ecosystem can and will develop around any water source. And it will adapt with slowly changing conditions.
Take a valley, put-on a geothermal vent and cover it with thousands feet of ice. You get something like lake Vostok. Under high pressure from the ice above, water stays liquid at -3°C. It may have been connected to the sea and is now sealed-off from the rest of the world for millions of years. Despite that, the bottom ice-layer sample reveals richness in fungi and bacteria. Some bacteria are known to reside in the guts of fishes. There are even some multicellular organisms.
Oxygen levels are high, but I don't relate that to photosynthesis, as the ice layer above is too thick. It may be conveyed by sliding glaciers from air naturally dissolved in the ice. Evidence of complex life is preliminary. Yet, it seems the essentials of life are there: oxygen, water and nutrients from geothetmal vents and water layers of varying salinity. You may build an ecosystem where a lake with complex life may have thrived on such energy and nutrients. The global weather gradually cause build-up of ice which shut-off the lake from the outside world. A geothermal vent may have prevented the ice on top to grow thicker (unlike Vostok) if you still want to let-in light to sustain algae and complex organisms.
Zeiss Ikon has a good answer but you also need to bear in mind that some lakes cannot ever become havens for complex aquatic life. For example if the country rock is limestone the water pH may consequently be too high to support the plant life that forms the base of the pond food-chain. Similarly a number of igneous rocks create acidic conditions that have the same effect for the opposite reason. If the hollow that the lake forms in has a lot of metal bearing ores the leached heavy metal salts can be extremely toxic. Some plants are adapted to such conditions but few animal species are. These chemical factors contribute to the relative sterility of many former quarries.