What exists (存在) in the sentence 孩子在家接受教育存在不足?
孩子在家接受教育存在不足。


It is a [topic] + [comment/ opinion sentence]

[孩子在家接受教育] (children receive education at home) is the topic

[存在不足] (insufficiency exist ) is the comment/ opinion

孩子在家接受教育存在不足 means "insufficiency exist in children home schooling "

To make it clear, let's expand the sentence 


不足處 (insufficient areas)
存在於 (exist in)
孩子在家接受教育 (children receive education at home)
的系統 (the system of)
之中 within


不足處存在於孩子在家接受教育的系統之中 = insufficient areas exist in the system of children receive education at home (home schooling children)

Back to the original sentence

存在不足 = insufficiency exist/  have insufficiency

存在 (exist/ have) is a verb here

It is an odd way to use 存在 without a preposition in a Chinese sentence. 

The typical way of using  存在 is with a preposition

Example:


[不足處 (subject)] [存在 (verb)][於 (preposition)] [孩子在家接受教育的系統 object][之中 (preposition)]
[雙方(subject)] [之間 (preposition)][存在著 (verb in continuous/ progressive tense)] [矛盾 (object)]