So "they manage to write" (it was unlikely that they could but they have somehow been able to) and "they are successful in writing" (they excel in writing, their skills or acheivments are excellent) are very different meanings. Which one does this sentence mean?
Succeeding to write doesn't work in english. Succeed doesn't go with to. They are succeeding at writing, they succeed at writing, they are managing to write, or they manage to write are all acceptable, though.