iex(1)> data = [["\"firstname\"", "\"lastname\""], ["\"Foo\"", "\"Bar\""]]
[["\"firstname\"", "\"lastname\""], ["\"Foo\"", "\"Bar\""]]
iex(2)> IO.iodata_to_binary MyParser.dump_to_iodata(data)
"\"\"\"firstname\"\"\";\"\"\"lastname\"\"\"\n\"\"\"Foo\"\"\";\"\"\"Bar\"\"\"\n"
iex(3)> 

… so not exactly.

rfc4180

5. Each field may or may not be enclosed in double quotes (however
   some programs, such as Microsoft Excel, do not use double quotes
   at all). If fields are not enclosed with double quotes, then
   double quotes may not appear inside the fields.

NimbleCSV has no problem consuming

"\"firstname\";\"lastname\"\n\"Foo\";\"Bar\""

it just doesn’t seem interested in producing it (likely to avoid unnecessarily bloating the output).

iex(1)> MyParser.parse_string("\"firstname\";\"lastname\"\n\"Foo\";\"Bar\"",[headers: false])
[["firstname", "lastname"], ["Foo", "Bar"]]
iex(2)> 

6. Fields containing line breaks (CRLF), double quotes, and commas
   should be enclosed in double-quotes.

NimbleCSV will only use the double quotes when absolutely necessary, e.g.:

iex(1)> data = [["firstname", "lastname"], ["Fo\no", "Bar"]]
[["firstname", "lastname"], ["Fo\no", "Bar"]]
iex(2)> IO.iodata_to_binary MyParser.dump_to_iodata(data)   
"firstname;lastname\n\"Fo\no\";Bar\n"
iex(3)> data = [["firstname", "lastname"], ["Fo\"o", "Bar"]]
[["firstname", "lastname"], ["Fo\"o", "Bar"]]
iex(4)> IO.iodata_to_binary MyParser.dump_to_iodata(data)   
"firstname;lastname\n\"Fo\"\"o\";Bar\n"
iex(5)> 

that last example also demonstrating

7. If double-quotes are used to enclose fields, then a double-quote
   appearing inside a field must be escaped by preceding it with
   another double quote.