To continue a string on a new line you need to concatenate.
ws.Cells(irow, 21).Formula = "=IF($J" & irow & "=""-"",IF($X" & irow & "=""-"",""No Data"",IF($X" & irow & "<$A$3,""Expired"",IF($X" & irow & "<$A$4,IF(OR(ISNUMBER(SEARCH(""O"",$W" & irow & ")),ISNUMBER(SEARCH(""P"",$W" & irow & "))),IF(ISNUMBER(SEARCH(""RMDL"",$W" & irow & ")),""(R) ATV & IRB"",""(R) ATV Only""),IF(ISNUMBER(SEARCH(""RMDL"",$W" & irow & ")),""(R) IRB Only"",""Invalid"")),IF(OR(ISNUMBER(SEARCH(""O"",$W" & irow & ")),ISNUMBER(SEARCH(""P"",$W" & irow & "))),IF(ISNUMBER(SEARCH(""RMDL"",$W" & irow & ")),""ATV & IRB"",""ATV Only""),IF(ISNUMBER(SEARCH(""RMDL"",$W" & irow & ")),""IRB Only"",""Invalid"")))))" & _
            ",IF($M" & irow & "<16,""Underage"",IF($X" & irow & "=""-"",""No Data"",IF($X" & irow & "<$A$3,""Expired"",IF($X" & irow & "<$A$4,IF(OR(ISNUMBER(SEARCH(""O"",$W" & irow & ")),ISNUMBER(SEARCH(""P"",$W" & irow & "))),IF(ISNUMBER(SEARCH(""RMDL"",$W" & irow & ")),""(R) ATV & IRB"",""(R) ATV Only""),IF(ISNUMBER(SEARCH(""RMDL"",$W" & irow & ")),""(R) IRB Only"",""Invalid"")),IF(OR(ISNUMBER(SEARCH(""O"",$W" & irow & ")),ISNUMBER(SEARCH(""P"",$W" & irow & "))),IF(ISNUMBER(SEARCH(""RMDL"",$W" & irow & ")),""ATV & IRB"",""ATV Only""),IF(ISNUMBER(SEARCH(""RMDL"",$W" & irow & ")),""IRB Only"",""Invalid"")))))))"