me parece que por esta pregunta hay dos interpretationes, no?

Hola Frank,

The typical phrase is "¿me pone un cafe por favor?" and is a request using the usted conjugation, since the server would probably be a stranger, or your relationship with him or her is in a formal setting. If you had a familiar relationship with the server perhaps a family member or friend, "me pones un café" would be fine. Either way it translates literally as "¿[will you] put [for] me a cup of coffee {on the table} please?" 

¿"Me pongo un café por favor?" doesn't work because it suggest a pronominal construction of the verb poner, which would be "Ponerse." The problem is that "ponerse" doesn't actually mean "to put or to place". Ponerse + Algo (something) would typically mean to put something on yourself; for example "El niño se pone su camiseta" means the boy puts on his T shirt. 

Just to be clear, if you are wondering about using "¿Yo pongo un café por favor?" it won't work either, because it translates as "I put a coffee please." So the constructions in the first paragraph above is how you ask for a coffee. All of this will become easier when you learn more about direct and indirect objects, and pronominal verb constructions. 

I hope this helps. Saludos. John