Завтра я подам заявление на отпуск в отдел кадров.

Подам is the 1st person singular future form of the verb подать.

• подать = a perfective verb, meaning to submit / hand in
• Perfective verbs do not have a present tense in the usual sense
• Their “present-looking” forms actually refer to the future

So:

• я подаю = I am submitting / I submit regularly (imperfective)
• я подам = I will submit / I will hand in (perfective, one completed action)

In this sentence, the speaker means a single completed action tomorrow, so подам is the natural choice.

Because подать and дать are not the same in meaning.

• дать = to give
• подать = to submit, hand in, file

When talking about documents such as:

• заявление = application / formal request
• документы = documents
• жалобу = complaint

Russian very often uses подать.

So подам заявление means I will submit an application, not simply I will give an application.

Yes, я could be omitted.

Russian often drops subject pronouns when the verb already makes the subject clear. Since подам clearly means I will submit, the sentence could also be:

Завтра подам заявление на отпуск в отдел кадров.

That sounds perfectly natural.

Including я can do things like:

• add clarity
• add emphasis
• contrast with someone else

For example, Завтра я подам... may sound a little more deliberate: Tomorrow I will submit...

It is in the accusative case, because it is the direct object of подам.

The dictionary form заявление is nominative singular, but for neuter inanimate nouns, the accusative singular is the same as the nominative singular.

So:

• nominative: заявление
• accusative: заявление

That is why the form does not change here.

This is a very common Russian pattern: заявление на + accusative for an application/request connected with something.

So:

• заявление на отпуск = an application for leave / vacation
• literally something like an application for leave

The noun отпуск is in the accusative singular, but since отпуск is an inanimate masculine noun, its accusative is the same as the nominative:

• nominative: отпуск
• accusative: отпуск

This whole phrase is a set, very standard office/workplace language.

Not exactly.

Заявление is usually a formal written statement/request/application, especially in administrative or official contexts. In this sentence, it is the normal word for a formal leave request.

Заявка is more like:

• a request order
• a service request
• a booking/request form
• sometimes an application in a more practical/technical sense

So for taking leave from work, заявление на отпуск is the standard phrase, not usually заявка на отпуск.

Here в shows movement to / into a place or institution.

The speaker is submitting the application to the HR department, and Russian expresses this as motion toward that department:

• в отдел кадров = to the HR department / to the personnel office

After в with motion, you normally use the accusative case.

So:

• в отдел = into / to the department

This is very natural with verbs like:

• нести в офис = take to the office
• подать в суд = file in court
• отправить в отдел = send to the department

Because кадров is in the genitive plural.

The phrase отдел кадров literally means something like:

• department of personnel
• staff department

Here:

• отдел = department
• кадры = personnel / staff
• кадров = of personnel / of staff

Russian often uses this structure:

noun + genitive noun

Examples:

• дом брата = the brother’s house / house of the brother
• стакан воды = a glass of water
• отдел кадров = personnel department

So кадров is there because it depends on отдел.

Отдел is in the accusative singular because it follows в with the idea of movement/direction.

Compare:

• в отдел кадров = to the HR department (motion, accusative)
• в отделе кадров = in the HR department (location, prepositional)

This is a very important contrast in Russian:

• в + accusative = into / to
• в + prepositional = in / inside

So this sentence uses в отдел кадров because the application is being submitted to that department.

Yes. Russian word order is flexible, though not random. The basic meaning stays the same, but emphasis changes.

Original: Завтра я подам заявление на отпуск в отдел кадров.

Possible variations:

• Я завтра подам заявление на отпуск в отдел кадров.
• Заявление на отпуск я подам завтра в отдел кадров.
• В отдел кадров я завтра подам заявление на отпуск.

The original version sounds very natural because it starts with завтра, setting the time first: Tomorrow...

That is common when the time is important context.

A helpful stressed version is:

За́втра я пода́м заявле́ние на о́тпуск в отде́л ка́дров.

Main stresses:

• за́втра
• пода́м
• заявле́ние
• о́тпуск
• отде́л
• ка́дров

A learner may especially want to notice:

• пода́м — stress on the second syllable
• заявле́ние — stress on -ле́-
• отде́л — stress on the second syllable
• ка́дров — stress on the first syllable

Yes, it sounds fairly neutral-to-formal because of the workplace/administrative vocabulary:

• подать заявление
• на отпуск
• отдел кадров

This is exactly the kind of wording you would expect in a work or office context.

A more casual English translation might be Tomorrow I’ll put in a vacation request, but the Russian itself is standard and professional, not slangy.

Yes. Отпуск often gets translated as vacation or leave, depending on context.

In many work contexts, отпуск means a period of approved time off from work. In English, that could be:

• vacation
• annual leave
• leave

So заявление на отпуск can be understood as:

• application for leave
• vacation request
• leave application

The best English wording depends on the situation, but the Russian phrase itself is standard.