「[私]{わたし}じゃ[歌]{うた}えない。」
=
「私では歌えない。」
「じゃ」 is the colloquial contraction of 「では」. More importantly, this 「では/じゃ」 placed directly after the subject of a sentence is quite nuanced. Therefore, the plain and neutral 「は」 cannot replace it for the nuance.
It means that the subject (「私」 in this case) is not good enough to do something for some reason.
"I could not sing (this song), (but some other people probably could)"
"I am not good enough to sing (this particular song)."

(source: smilevideo.jp)
「[僕]{ぼく}じゃダメですか?」 = "Am I not good enough for you?"