Таможенный представитель и почтовый оператор работают вместе.

Both noun phrases are in the nominative case, because they are the subject of the sentence:

• таможенный представитель
• почтовый оператор

The adjectives таможенный and почтовый also appear in the nominative, matching the nouns they describe.

вместе is an adverb, so case does not apply to it.

Because they are adjectives agreeing with masculine singular nouns in the nominative case.

• таможенный представитель — both words are masculine singular nominative
• почтовый оператор — same pattern

In Russian, adjectives must agree with the noun in gender, number, and case.

Both are masculine nouns.

A few clues:

• оператор ends in a consonant, which is a very common masculine pattern
• представитель ends in -ь, which can be masculine or feminine, so you need to learn its gender as part of the word; представитель is masculine

You can also see the masculine agreement in the adjectives:

• таможенный representative
• почтовый operator

If the nouns were feminine, the adjectives would look different.

Because the subject is actually two people joined by и:

• таможенный представитель
• и
• почтовый оператор

Together they form a compound subject, so the verb is plural:

• работают = they work / are working

This is the same idea as in English: The representative and the operator work together, not works together.

The dictionary form is работать — to work.

работают is the 3rd person plural present-tense form, meaning:

• they work
• they are working

So here it agrees with the two-person subject.

It can mean either one.

Russian does not have a separate present continuous form like English are working. The present tense often covers both:

• работают = work
• работают = are working

Which one sounds best in English depends on context.

Вместе means together. It is an adverb.

It describes how they work:

• работают вместе = work together

It does not change form.

The given order is the most neutral and straightforward:

• Таможенный представитель и почтовый оператор работают вместе.

Russian word order is more flexible than English, so other orders are possible for emphasis, for example:

• Вместе работают таможенный представитель и почтовый оператор.

But the original sentence is the best neutral version for a basic statement.

Because работают already contains the full idea they work / they are working.

In English, present continuous needs are working. In Russian, the present-tense verb form by itself usually does the job:

• они работают = they work / they are working

So no extra word for are is needed here.

The stress is:

тамОженный представИтель и почтОвый оперАтор рабОтают вмЕсте

A rough pronunciation guide:

ta-MO-zhen-nyy pree-sta-VEE-tyel' ee pach-TO-vyy a-pye-RA-tar ra-BO-ta-yut VMYE-stye

The stressed syllables are important in Russian, so it helps to memorize words together with stress.