Do any organisms recycle their own waste internally?
All organisms recycle their waste internally. Every cell of every living organisms is constantly breaking things down and re-using the components so produced. But you're presumably wondering about things such as carbon dioxide, urine and faeces? These are not recycled because the benefits of doing so are not worth the costs.

Let's consider carbon dioxide as an example. We know that carbon dioxide can be usefully converted into energy-holding molecules using readily available sources of water, oxygen and sunlight. So why do animals** "wastefully" breathe it out? Because the energetic benefits of respiration are high compared to the energetic benefits of photosynthesis and so photosynthesis is unable to make a useful contribution to the overall energy balance of an animal while the costs of the effectively photosynthesising are quite high (e.g. having leaf-like protrusions, synthesizing chlorophyll and so on).

In fact, it must always be true that there are processes which are not energetically worthwhile because of entropy. An organism that could perpetually recycle its waste would be a perpetual motion machine.

** - and many microbes and, in fact, plants - although their intake of carbon dioxide is typically higher over the solar cycle.