The reason that there is no option to select Contact when adding fields to a Requirement form is that it is built this way by design. Contact records tie to the Organization and Request level, but not directly to the Requirement level records. We understand the need for this sort of feature and have created an Idea on our Community page to this effect. Please navigate to and log in or create an account. Once logged in, you may navigate to the Idea page below in order to comment and vote on the Idea:

Remember, the more votes an Idea receives, the more traction it will build with our Product Managers, so please encourage your colleagues to vote on the Idea as well!