Is it permissible for me to pray at home if I am forced to live in an area whose people are innovators (ahl al-bid'ah), and the nearest mosque is 700 meters away, and I do not own a car?

According to the view that congregational prayer is obligatory for individuals, the criterion for its obligation is hearing the voice of the muezzin without a loudspeaker under normal circumstances. Thus, whoever hears the call must go to the mosque, even if walking. As for someone who does not hear it, congregational prayer is not obligatory for them, even if they own a car.

As for praying behind the innovators mentioned in the question, it is not valid; such as one who seeks aid from other than Allah, or supplicates to other than Him, or slaughters for other than Allah, or believes in the deficiency of the Qur'an, or slanders the honor of the Mother of the Believers A'isha, or exaggerates in devotion to Ali and invokes him besides Allah, or insults the Companions.