I am a 16 year old mother and do not wish to live with my mother and step father any longer. My grandmother and grandfather want to take me and my son in because they know of my living circumstances. Is it possible for me to be emancipated if I am not financially responsible but my grandparents want to be and would be over me and my son?

You may want to review the requirements for emancipation in Arizona. The requirments for emancipation are listed in the laws section of this site. Note the following portion of that section:

Minor must demonstrate to the court the ability to: 

• Manage financial, personal and social affairs.
• Live wholly independent of parent or guardian.
• Obtain or maintain health care, education, training or employment.