Hi Mike,
Might be easier to manipulate by parsing the HTML as XML e.g.
xml = ImportString[html, {"HTML", "XMLObject"}];
Cases[xml, XMLElement["label", l___] :> l, Infinity]
(*
{{"for" -> "field_1"}, {"Field 1"}, {"for" -> "field_2"}, {"Field 2"}, {"for" -> "field_3"}, {"Field 3"},
{"for" -> "field_4"}, {"Field 4"}, {"for" -> "field_5"}, {"Field 5"}, {"for" -> "field_6"}, {"Field 6"},
{"for" -> "field_7"}, {"Field 7"}, {"for" -> "field_8"}, {"Field 8"}, {"for" -> "field_9"}, {"Field 9"}}
*)