How do congenitally deaf and mute people think? If a person is born deaf and dumb, how can they think? In "what language" do these people think? Do they develop their own inner language?

Unfortunately I have not found an answer, and I actually doubt it has ever been elucidated.
Short answer
The inner voice of congenitally (pre-lingually) deaf people who have not received treatment like cochlear implantation, is not sound-based. Instead, it is mainly based on visual images, such as sign-language or printed material.  

Background
According to an anecdotal report published in the Independent of a congenitally deaf person, who deliberately refused cochlear implantation or other treatments, the inner voice is a visual entity taking the form of sign-language, or visual images, or sometimes printed words. 

An interesting report from UCL investigated congenitally deaf people suffering from positive psychotic symptoms. In normal-hearing people, positive psychotic symptoms typically involve hearing voices (auditory hallucinations). In congenitally deaf folks, who obviously never had the chance to hear any voices in their lives, these auditory hallucinations were described as


  [Not consisting of] sounds, but [...] the gender and identity of the voice [were recognized. They were] image[s] of [...] voice[s] signing [,] or lips moving in their mind.


Source
- The Independent, December 21, 2013, UK