。
She sang the song while crying.
Literal
She [topic-は] tears [object-を] shedding [while-ながら] that song [object-を] sang.
Note the double を — 涙を流しながら and その歌を歌った each have their own object, which is natural because the ながら clause effectively functions as an adverbial modifier. 涙を流す (なみだをながす, 'to shed tears') is a set expression — more literary than 泣く ('to cry').