Both pedir and preguntar mean "to ask", however they are used in different contexts. Pedir is generally used when asking for something or when ordering food at a restaurant. Preguntar is simply used to ask questions.
Examples:
"For my birthday, I asked for a new phone."
Para mi cumpleaños, pedí un teléfono nuevo.
"I ordered the fish dish.'
Pedí el plato de pescado.
"She asked the professor when the assignment is due."
Ella le preguntó al profesor para cuándo es la tarea.
The sentence is "I ask for a new dress", therefore the use of pedir is correct. The subject is "I", thus the correct conjugation is pido.
Answer:
Pido un vestido nuevo.