The correct option is
B are going
In order to fill in the blank with the most appropriate helping verb, we must understand the 'subject and verb agreement'. A subject is a person, thing, or a concept, in a sentence. A sentence always describes of explains us about a person, thing or a concept and that is called 'subject' of the sentence. The rest of the sentence is a complement (which has the object in it). A subject and a verb should always agree in a sentence. If the subject is singular (example: Raj), the verb will also be singular (does). If the subject is plural (We), the verb will also be plural (do).As the given sentence asks us to fill in the blank with a verb that agrees with the subject and not with the complement, we will focus on the subject only.
NOTE: When 'or' is used in a sentence, the subject becomes singular. But, if the second subject is plural, the verb will also be plural. In the given sentence, the second subject 'brothers' is a plural subject, hence the verb will be plural as well.
Option A - 'is' is a singular subject. As we need a plural verb, we can't use 'is going' here. Hence, option A is incorrect.
Option C - 'Will be going' is used to represent an action of going somewhere in future continuous tense. Here, the action is not 'going' it is 'win'. Hence, option C is incorrect.
Option D - 'Are' is a plural verb and can be used with a main verb. As 'are' is a helping verb, it can't be used with an infinitive with 'to'. Hence, option D is incorrect.
Option B - 'Are going' is a plural helping verb with a main verb in participle form. As the subject is plural, we can use a plural verb. Hence, option B is the correct answer.
'Shilpa or her brothers are going to win the prize.'