Заявку на участие нужно подать заранее.

Because заявку is in the accusative case, which is used here for the direct object of the verb подать.

• Dictionary form: заявка = application, request
• Accusative singular: заявку

In this sentence, the action is подать что? = to submit what?
Answer: заявку

So:

• заявка = the application (subject form / dictionary form)
• заявку = the application (object form)

На участие uses the preposition на with the accusative case.

• участие is a neuter noun
• Its nominative and accusative singular forms are the same: участие

So even though the form does not change visibly, it is still accusative here.

The phrase заявка на участие means an application to participate or an application for participation.

This is just a common noun pattern in Russian:

• заявка на участие
• билет на концерт
• запись на курсы

The preposition на often shows the purpose or goal of something.

Нужно means something like it is necessary, one must, or it’s needed to.

Russian often uses impersonal constructions, where there is no explicit subject. English usually wants a subject, but Russian does not always need one.

So:

• Нужно подать заявку = It is necessary to submit an application
• Заявку нужно подать заранее = The application needs to be submitted in advance

This structure is very common:

• нужно сделать
• надо прийти
• можно войти
• нельзя курить

It focuses on necessity, permission, or prohibition rather than on who does the action.

Подать is the perfective form, while подавать is the imperfective form.

Here, the sentence is about completing a single action: submitting the application. Russian often uses the perfective infinitive after нужно when the idea is you need to do it / complete it.

• подать = to submit, to hand in (as a completed act)
• подавать = to be submitting / to submit repeatedly / to submit in general process terms

So:

• Нужно подать заявку = you need to submit the application
• Нужно подавать заявки заранее could mean applications need to be submitted in advance in general, or repeatedly, depending on context

In this sentence, a specific required action is meant, so подать is the natural choice.

Подать is very common for officially submitting documents, forms, applications, requests, etc.

Typical combinations:

• подать заявку = submit an application
• подать заявление = submit a formal statement/application
• подать документы = submit documents

By contrast:

• отправить = send
• послать = send

Those focus more on sending something from one place to another, while подать often suggests submitting formally to an institution, office, committee, website, and so on.

So подать заявку is the standard phrase.

Заранее means in advance, ahead of time, or beforehand.

It is an adverb, so it describes how/when the action should be done.

In this sentence:

• Заявку на участие нужно подать заранее.

This is a very natural position, at the end of the sentence.

But Russian word order is flexible, so you can also say:

• Заранее нужно подать заявку на участие.
• Нужно заранее подать заявку на участие.

All of these are grammatical, but the emphasis changes slightly.

• End position often sounds neutral and smooth
• Earlier position can emphasize the time factor more strongly

Russian word order is more flexible than English word order because case endings help show grammatical roles.

Starting with Заявку на участие puts that phrase in focus as the topic:

• Заявку на участие нужно подать заранее.

This can feel like: As for the application to participate, it needs to be submitted in advance.

A more neutral version is also possible:

• Нужно подать заявку на участие заранее.

Both are correct. The version with заявку first gives a little more prominence to the application itself.

Very close, yes. In this sentence, нужно and надо are often interchangeable.

• Заявку на участие нужно подать заранее.
• Заявку на участие надо подать заранее.

Both mean essentially the same thing: the application needs to be submitted in advance.

Very roughly:

• надо can sound slightly more conversational
• нужно can sound slightly more neutral or formal

But in many real situations, the difference is small.

It can naturally be understood both ways in English, because the Russian sentence is impersonal and does not name the doer.

Russian focuses on the necessity of the action:

• нужно подать = it is necessary to submit

So depending on context, you might translate it as:

• You need to submit the application in advance
• The application must be submitted in advance
• One must submit the application in advance

Russian often leaves the agent unstated when it is obvious, general, or not important.

Yes, it is very useful to learn it as a set phrase.

• заявка = application
• на участие = for participation / to participate

Together:

• заявка на участие = application for participation

Russian often builds expressions like this with a noun + на + another noun:

• запрос на информацию = request for information
• разрешение на работу = work permit / permission to work
• билет на поезд = ticket for the train

Learning common combinations as chunks will make your Russian sound more natural.

Sometimes, but not always. Заявка and заявление are related but not identical.

• заявка often means an application, request, order, booking, or registration-type submission
• заявление is usually a more formal written statement or official application

For joining an event, competition, program, or participation list, заявка на участие is extremely common and natural.

So in this sentence, заявка is the best choice.

A few stress points are useful:

• зая́вку
• уча́стие
• ну́жно
• пода́ть
• зара́нее

The letter о in unstressed positions is often pronounced closer to a-like sounds, so beginners may notice that spoken Russian does not always match the spelling exactly.

A natural rhythm would be something like:

Зая́вку на уча́стие ну́жно пода́ть зара́нее.

You could add the person who must do it, or change the structure entirely.

Examples:

• Вам нужно подать заявку на участие заранее.
  = You need to submit an application in advance.

• Участникам нужно подать заявку заранее.
  = Participants need to submit an application in advance.

• Вы должны подать заявку на участие заранее.
  = You must submit an application in advance.

Compared with нужно, должны sounds stronger and more direct.