Learn Croatian: Poštovani, javljam se na vaš natječaj. - Dear Sir/Madam, I am applying to your job advertisement.

Poštovani literally translates to "Respected." It is a masculine plural adjective used as a noun, and it is the standard, safe opening for formal letters and emails in Croatian when you do not know who exactly will read your message.

Does the verb javiti se specifically mean "to apply" for a job?

No, the phrase javljam se literally means "I am getting in touch" or "I am reporting myself." However, when combined with na natječaj (to a job competition or advertisement), it is the most natural way to say "I am applying" in Croatian.

Which case is used for vaš natječaj after the preposition na?

It is in the accusative case. With the verb javiti se, the preposition na points to the target of your action, requiring the accusative. Because natječaj is a masculine inanimate noun, its accusative form looks identical to the nominative dictionary form.

Why does the reflexive pronoun se come immediately after javljam?

The word se is an enclitic, meaning it is a weak word that strictly requires the second position in a clause. The comma after Poštovani resets the word order. Therefore, javljam is the first word of the new clause, and se safely takes the second spot.