sinseisyo ni kinyuusuru.

申請書 means application form or application document.

• 申請 = application, request, filing
• 書 = document, written form

So 申請書 is the form or paperwork used to apply for something.

Here, に marks the place or target where information is written.

So:

• 申請書に記入する = write/fill in information on the application form

With 記入する, に often shows the form, box, line, or document where something is entered.

Examples:

• 用紙に記入する = fill something in on a form
• 欄に記入する = fill something in in a box/blank

Actually, 申請書を記入する is also possible, but the nuance is a little different.

• 申請書に記入する focuses on writing information onto the application form
• 申請書を記入する treats the whole form as the thing being filled out

A useful way to think about it:

• に = where you write
• を = what you fill out

So both can appear, depending on what the speaker wants to emphasize.

Compare:

• 名前を申請書に記入する = write your name on the application form
• 申請書を記入する = fill out the application form

記入する means to fill in or to enter information in a form, document, or blank space. It is more formal and more specific than 書く.

• 書く = to write
• 記入する = to fill in, enter required information

So if you are talking about forms, questionnaires, applications, or blanks, 記入する is very natural.

Examples:

• 名前を書く = write your name
• 名前を記入する = enter your name (on a form)

In many administrative or office situations, 記入する sounds more appropriate.

Yes. Japanese often omits words that are understood from context.

In 申請書に記入する, what is being written is not stated. A fuller sentence might be:

• 名前を申請書に記入する。 = Write your name on the application form.
• 必要事項を申請書に記入する。 = Fill in the required information on the application form.

So the sentence as given is a shortened expression, especially natural in instructions or notes.

The dictionary form is often used in:

• instructions
• manuals
• checklists
• labels
• headings
• brief written directions

So 申請書に記入する。 can feel like a short instruction such as Fill in the application form.

A more directly polite instruction would be:

• 申請書に記入してください。

Both can be used, but the dictionary form is common in concise written instructions.

No subject is stated, but Japanese often leaves the subject out when it is obvious from context.

Depending on the situation, the implied subject could be:

• you in an instruction
• I/we in a statement about what someone will do
• some other understood person

For example, on a sign or instruction sheet, the implied meaning is usually:

• (Please) fill in the application form.

It is pronounced:

しんせいしょ に きにゅうする

Breakdown:

• 申請書 = しんせいしょ
• に = に
• 記入する = きにゅうする

Yes, in context it often functions like an instruction, even though it is not a direct command form.

For example, on paperwork or in a manual:

• 申請書に記入する。

can effectively mean:

• Fill in the application form.

Japanese often uses plain forms in written instructions where English might use the imperative.

Some very common fuller versions are:

• 申請書に名前を記入する。
  Fill in your name on the application form.

• 申請書に必要事項を記入する。
  Fill in the required information on the application form.

• 申請書に住所と電話番号を記入する。
  Fill in your address and phone number on the application form.

These make it clearer what information is being entered.