Is there any neutral element for the cryptographic hash function SHA256? (or its variants)
No one knows.  It is believed to be computationally infeasible to find such a string, if one exists.  No one knows whether such a string exists.
By standard heuristics (modelling SHA256 as a random function), there is approximately a $1/e\approx 0.368$ chance that no such input exists, and approximately a $0.632$ chance that such an input exists, so if I were forced to bet, I would bet that it's more likely there exists such an input than that there doesn't, but no one knows.  Don't interpret this chance too seriously: for the specific SHA256 function, either such an input exists or it doesn't.  We don't know which is the case.
If such an input is found, it is believed that it would take about $2^{256}$ steps of computation to find it, which is completely infeasible.