The correct option is B On going home, I found my dog missing.
The words in the bracket is a group of prepositions; while combining a
sentence with a preposition, it is often followed by a noun or a gerund.
When the given sentences are combined, a gerund/noun follows the preposition.
In option B, the preposition 'on' is followed by the gerund 'going' and noun 'home' which replaces the statement "I went home.", 'the verb 'went' is changed to a gerund form 'going'. It is the correct form, hence B is right.
Option A has omitted the preposition 'on', whereas the question asks for it to be used, hence A is incorrect.
Option C has changed its meaning from the original sentence, 'going on' means 'to carry forward', whereas the required phrase is 'on going', which means 'once I reached home'. Hence C is incorrect.
Option D has also changed in meaning (It gives the impression that the dog misses going home.),also, "dog missing going home" is grammatically incorrect.