You can say it either way.
If each person in that certain number of people needs to write either one letter or multiple letters, you can say:
You will need to have a certain number of people write letters of
While the above is correct when you need only one letter from each person, if you want to explicate the fact that you only need one letter from each person, it's also completely proper to say:
You will need to have a certain number of people write a letter of
The caveat, of course, is to take care not to say it this way when it might be construed as the group writing a single letter.
Saying "letter" instead of "letters" is the same grammatical rule a teacher follows when she instructs her class of first-timers, "Students, raise your hand whenever you have a question." If the teacher were to say, "Students raise your hands when you have a question," these newbies could easily misunderstand that they're supposed to raise both of their hands. The teacher saying "hand" instead of "hands" eliminates any confusion that her using the plural might cause while it has little to no risk of her students misunderstanding her to mean that they all collectively possess a single hand that they are to raise.
In the end, the speaker must weigh the situation and decide which way is preferable. The rule of grammar on this point is anything but hard-and-fast. For more on using a singular noun object with plural noun subjects, see the following on subject-compliment agreement: