There are several reasons for not seeing the "Request stud" link on a stallion.

1. The stallion is not level 5
2. The stallion is not yet 3 years old or is retired.
3. The stallion is starving
4. The stallion does not have the vet checks (breeding check, the regular check and does not have the shots)
5. The stallion is 'currently busy', meaning that it will have to wait for several days.

Note that mules cannot breed.