、。
The moment she saw me, she burst into tears.
Literal
She [topic-は] me [object-を] sees [the instant-や否や], suddenly-loudly began-crying.
~や否や is a literary, formal conjunction meaning 'the instant X, Y' — adjacent to ~たとたんに and ~なり in meaning but with a more bookish, almost narrative-prose register (the 否や is classical, literally 'whether or not,' giving it that dramatic 'no sooner had X happened than Y' feel). わっと is a sound-mimetic specifically for sudden loud sobbing or shouting — sharp and immediate. 泣き出す is a compound verb pairing 泣く ('cry') with the sudden-onset auxiliary ~出す ('burst into') — 'burst into tears.' The whole sentence is heightened narrative, the kind you'd see in a novel.