If the question is are Darlingtonia self-compatible - meaning that a plant can be fertilized by pollen from the same plant the answer is yes. Self-incompatible plants need pollen from a genetically different plant.
If the question is will the pollen get placed on the stigma when the flower closes the answer is no since the flower doesn't close and press the anthers against the stigma.
That doesn't mean an animal of some sort (bird, insect, spider, human etc.) or even the wind will transfer pollen to the stigma.
The design of the flowers appear to encourage cross pollination. The bell shaped ovary might shield pollen from dropping on to the stigma. The placement of the "entry ports" on the petals and the bell shaped ovary might make it easier for an insect entering the flower to transfer pollen. An insect that was gathering pollen from the anthers after having moved past the "bottleneck" caused by the flare of the ovary may have a higher tendency to drop out the bottom opening of the petals. I seem to recall field observations that the insects tend to drop out the bottom rather than climb back of the "entry ports". See photos below.