Every one of the support staff and the Participants have to

Each/Every changes the verb into singular (i.e. overrides the number rule for conjunction)
For example: Every boy and girl is required to wear an uniform.

We have a compound subject here--"Every one of the support staff and the participants"--so we can eliminate "has" in D & E.

A & C have a problem with the placement of the modifier "compulsorily." Saying that they "have to compulsorily go" is redundant, since "compulsorily" refers to something you have to do. They don't have to "compulsorily go"; they just have to go. Note that C also incorrectly uses "he or she" to refer back to our plural subject.

I'm assuming that B was supposed to have the adjective form "compulsory." The adjective modifies the noun phrase "full body search." I still feel that this is redundant, but at least it doesn't incorrectly modify the action. With this correction, B is the best choice.
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The answer choices are correct.

OA is D and the explanation says that "Each" is singular that's why HAS....but I know the subject is Plural (Every one of xyz AND participants)...not sure what's going on here

You know, it's funny--I didn't like that compound subject in B, and I originally figured the answer had to be D or E. If you read it the way the author seems to have intended, maybe it does work better: "Every one of (the support staff and the participants) has to . . ."

However, "has to compulsorily go" is absolutely wrong. I'd say we have a flawed question here. If B really says "compulsorily," there is no good answer.
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@Dmitry.
I agree. Why do we need to say "has to compulsorily go". As far as I know, "has to" means "compulsory". I doubt that "compulsorily" is redundant? Could you confirm please. Thanks.

Pqhai, when you say "I doubt that 'compulsorily' is redundant?" I'm assuming you mean that you think it is redundant. Watch your usage--"I doubt that X" means that you think X is unlikely to be true.

In any case, yes, as I stated in my initial response, "has to compulsorily go" is definitely redundant. Basically, the sentence is saying that people "have to have to go." I'm not even sure what that would mean . . .
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Im a bit unsure why is wrong to say: has to compulsory go............compulsory is also mandatory..

The subject has to go (mandatory or compulsory https://www.wordreference.com/definition/compulsory) towards something..........

For instance in my native language is perfectly fine from a boss or a director to say : You have to do this-----it is mandatory or compulsory
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First, the subject of this sentence is "Every one", not "the support staff and the Participants". The "of the support staff and the Participants" is a prepositional modifier of the subject. Therefore, we choose "has" instead of "have". A, B, and C are out.

Second, one should almost always put Gerund after a preposition (here: "before"). (I don't know exceptions to the rule - if you do, please name them.) Therefore, we choose "entering" instead of "they enter". "they" is also incongruent with "has". E is out.

D is correct.

You're right that if we're going to have a verb after a preposition, we need to turn it into a gerund. For instance, we can't say "Wash your hands before eat" or "I am interested in to sign up." Those are terrible!

However, there is no rule that we can't have a clause after a preposition, as we see in E. A few examples:

I would like to talk to you before you leave. (Notice that if we used the gerund here--"before leaving"--it would change the meaning of the sentence by making *me* the one who's leaving.)
I'm not going to go to Mars until there is a way to get back home. (There is no way to express this with a gerund after the preposition.)

This kind of construction doesn't work with all prepositions--I can't think of a way to follow basic prepositions such as "in" or "of" with a clause (although they can of course be followed by a noun). However, when we're using a word like "before" to show the relationship between two events, a clause is often needed.

So we can't eliminate based on the use of a clause after a preposition, but E is still wrong because of the pronoun "they."
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Well: the following six words can act as both a sub-conjunction and a preposition. If it is a preposition, then a noun follows. If it is acting as sub-conjunction, an IC with a full-fledged verb follows. Those six words are

1. After 2 Before 3. Until 4 Since 5. For 6. As

Let us see each of them:

After: I slept after my dinner (preposition) – dinner is a noun; it can also be a gerund as in I slept after reading
I slept after I felt tired: here after is a sub-conjunction followed by a verbed clause.

Before: I jogged before my bath – preposition
I jogged before I took my bath – conjunction

As: I am correct in my duties as a father --- preposition
I am correct in my duties as a father is required to do - Conjunction

Since: I slept for a good time since your arrival - preposition
I slept for a long time , since you arrived very late

Until: I slept late until 9 AM – preposition
I slept late until you arrived – conjunction

For: I slept for 12 hours- preposition
I slept the whole day, for I felt tired. Conjunction ( for used in the meaning of because)

So we have to see, whether the said word is used as a preposition or a conjunction, before taking a blanket decision

E is still wrong in the given case for other reasons.

HTH
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So let's put the rule together: Whenever a preposition is followed by verb, it has to take Gerund form. Right?

@HumptyDumpty

Again, not so blanketely:

A word that is a preposition never is followed by a verb; As long as it is a preposition, it can only be followed by a noun or noun phrases. Some of these nouns may be gerunds. All gerunds are nouns in essence. But all nouns are not gerunds (gerunds are verb +ing forms, with the name ‘verbal’)

If a word is subordinate-conjunction, what follow will be only a clause, meaning, and a clause with a verb;
Any attempt to short - cut this compass will only land us in unintended muddle. .
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Well, that is the meaning of the rule. Gerund is a verb form which functions as a noun.

Gerund is a present participle that functions as a noun. present participles are not verbs
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I used the definition from Consice Oxford English Dictionary, OUP 2007. I am not a pro, therefore I always draw on official resources.

Again, I want to stress that *all* of the answer choices are wrong. This is not a valid SC problem. However, the intended meaning is apparently for the subject to be "Every one," with "of the support staff and the participants" serving as a modifier. Therefore the plural pronoun "they" is incorrect. I think the use of "the" before "participants" is awkward and creates some ambiguity about the subject, which is why I initially said that the subject is plural. It's pretty safe to say that you wouldn't see this on the GMAT.
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Explain the reason for being D correct choice and others being wrong choice.

"Every one of" as a subject needs a singular verb, in this case "has", not "have. So eliminate A, B, and C as they have 'have' as the verb.

D and E differ in what comes after "before"; both "before entering" and "before they enter" are in fact correct.

"they" could be used to refer to indefinite pronouns, but it seems the test maker tried to avoid this (parallelism issue) by employing before + gerund, which makes D a little more correct (!).
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Can they, their, them, and themselves be used as singular pronouns?

They, their, them, themselves: English lacks a common-gender third person singular pronoun that can be used to refer to indefinite pronouns (such as everyone, anyone, someone). Writers and speakers have supplied this lack by using the plural pronouns. ⟨and every one to rest themselves betake — William Shakespeare⟩ ⟨I would have everybody marry if they can do it properly — Jane Austen⟩ ⟨it is too hideous for anyone in their senses to buy — W. H. Auden⟩ The plural pronouns have also been put to use as pronouns of indefinite number to refer to singular nouns that stand for many persons. ⟨'tis meet that some more audience than a mother, since nature makes them partial, should o'erhear the speech — William Shakespeare⟩ ⟨a person can't help their birth — W. M. Thackeray⟩ ⟨no man goes to battle to be killed. — But they do get killed — G. B. Shaw⟩ The use of they, their, them, and themselves as pronouns of indefinite gender and indefinite number is well established in speech and writing, even in literary and formal contexts. This gives you the option of using the plural pronouns where you think they sound best, and of using the singular pronouns (such as he, she, he or she, and their inflected forms) where you think they sound best.

quoted from merriam-webster

The explanation above given by tireks for A,B and C is correct. As for E, the pronoun "they" is plural - it has no plural antecedent: at least within the sphere of GMAT, "they" cannot refer to "every one".