We just use accusative when asking for things. You can consider that this sentence has the verb «дайте» 'give' dropped, if that helps.
I added the "give me a" part to the sentence. Should this be an accepted response?
No, that is not part of the sentence you're asked to translate in this case. It might still be considered correct in other situations, however.
If I were to request a cup, normally used for tea, would I use only 'Чашку' or would I have to identify it further in this question?
No. Cup is ча́шка, while glass is either стака́н (bigger glass, without a stem), бока́л (stem glass) оr рю́мка (smaller wine glass).
(Ча́шку is the accusative case of ча́шка. Accusative case of рю́мка is рю́мку. Бока́л and стака́н don’t change their form in the accusative case, they remain бока́л and стака́н.)