It’s good that you are thinking about a warm-up activity. Warmers are important to get students talking and thinking in English. To be effective, they need to be short and quick-paced.
The activities you suggest: a role play or arranging cut-up letters will take a lot of time; so I don’t suggest you use them as warmers. You can use them as the main focus of a lesson. Also, as you correctly point out, there won’t be a lot of speaking when students organise the letters.
Here are two ideas for warmers around the theme of letters. Of ocurse, you can change the theme, if you like. They are not too lively or physical, so they can suit a first lesson with business students.
1 Play a game of hangman using the word correspondence or communication. Students call out letters. When they find the word, ask them what it means and then discuss correspondence or oral and written communication. How often do students write letters? What type do they write in their own language? Do they prefer letters, phone, e-mail? What letters will / do they write in English? This discussion can be done in pairs, groups or as a whole class.
2 Find a picture of somebody writing a letter; a colour one from a magazine is best. Use A4 size and mount it on card so it is fairly stiff. Hold the picture towards you and flip it over quickly towards the students. They will only see it for a fraction of a second; so get their attention before doing it. Students tell you everything they see. At first, they will mention colours and very little else. You can then flash the picture again and students tell you what they saw. Continue until they have most elements or until students seem bored. Then show them the picture and talk about it. Discussion of letters can continue as above.