The cytokine storm is a hyperinflammatory state, due to excessive immune response.
It was a serious problem with earlier attempts to make coronavirus vaccines out of whole virus (in response to the SARS outbreak, over a decade ago). That technical approach was abandoned as inherently unsafe.
None of the authorized vaccines (none of which use whole coronavirus) have shown that risk - and it was specifically checked for in testing.
I believe you’re thinking of ADE.
Not what I meant.