。
She nearly fainted at the sight of blood.
Literal
She [topic-は] blood [object-を] saw on-the-verge-of-fainting became.
The construction ~そうになる ('come close to / nearly do') captures a near-event: the action almost happened. Built from the visual evidential ~そう (verb stem + そう = 'about to / on the verge of') plus になる ('become / come about'), it delivers 'reached the point of nearly X.' Useful for dramatic almost-misses: 倒れそうになった ('almost fell over'), 泣きそうになった ('almost cried'). Without そうになった the verb 卒倒する simply means 'fainted'; layering on そうになった pulls the climax back to a near-miss.