Here is an example of doing this by splitting to String array. Changed lines are marked as comments.
// Start of your code.
row = new Vector(); int i=0;
String[] st = lineData.split(","); // Changed
for (String s : st) { // Changed
row.addElement(s); // Changed
if (row.get(i).toString().startsWith("\"") == true) {
while (row.get(i).toString().endsWith("\"") == false) {
p1 = row.get(i).toString();
p2 = s.toString(); // Changed
row.set(i, p1 + ", " + p2);
}
...// Rest of Code here
}