I have ordered for a cup of tea
Or
I have ordered a cup of tea
Which one is correct and why?
You order something for someone or something. You don't order for something for someone or something in English. That doesn't sound idiomatic. For example:
I ordered a cup of tea for you.
I ordered a printer and twenty reams of A4 paper for the office.
I ordered for a cup of tea sounds like you placed an order for a cup of tea as though it were a living thing that was interested in receiving something from you, which is really just pure nonsense.
However, note that if the word order is used as a noun, then you most certainly can use it with the preposition for:
We received an order for a cup of tea.
I placed an order for a printer and twenty reams of A4 paper.