Setting the row terminator in addition to the field terminator should do the trick
'bcp DBName..vieter out c:\test003.txt -c -T -t"\",\"" -r"\"\n\"" -S SERVER'
This will likely work, but miss of the leading " for the first field of the first line, and perhaps the last of the last, i'm not sure, just guessing really, no server here!
or try using QUOTENAME to wrap text fields (you could also wrap numbers, but that isn't normally required.
'bcp "SELECT id, age, QUOTENAME(name,'"') FROM DBName..vieter" queryout c:\test003.txt -c -T -t"," -S SERVER'