To my knowledge there is no specific book of duas written by the 12th Imam (ajt). What we do have is a series of letters ( tawqi'at) attributed to the Imam many of which which were written by him or dictated by him. As for dua's there are numerous ones that are attributed to the Imam and can be found in early sources such as Misbah al-Mutahajjid of Shaykh Tusi ( d.460 Ah). Sh al-Tusi had numerous turuq or isnads reaching contemporaries of the Imam during ghayba al-sughra as did later scholars such as Ibn al-Mashhadi who narrates the famous Ziyarat Al Yasin which was taught by the 12th Imam when we wish to turn to him and seek his guidance. To conclude, there are numerous supplications and even some ziyarat which can be reasonably traced back to the 12th Imam. However there is no book written by him per say at least in any extant surviving form.
wa Allahu al-'alim,
And God is the knowing,