If it's exactly that, then you can replace runs of "," with "\n" and that will only match those commas between quotes rather than inside them.
Code:
:cat data
{"field(1)":"value(1)","field(2)":"value(2)","field(n)":"value(n)","field(z)":"comma,stuff"}
:sed -e '1s/^.//' -e '$s/.$//' -e 's/","/"\n"/g' data
"field(1)":"value(1)"
"field(2)":"value(2)"
"field(n)":"value(n)"
"field(z)":"comma,stuff"
Of course, if the data isn't that exact, then that may not work.