Igazoljuk az állampolgárságot, különben nem tudják kiállítani a magánútlevelet.

It can be either, because igazoljuk is one of those Hungarian forms that can look the same in:

• present indicative: we verify / we prove it
• imperative / exhortative: let’s verify / let’s prove it

So the exact meaning depends on context and tone.

In a sentence like this, let’s verify the citizenship, otherwise they can’t issue the private passport is often the more natural reading, because különben gives a consequence and makes the first clause sound like a necessary action.

Because állampolgárságot is the direct object of the verb igazoljuk.

• állampolgárság = citizenship
• állampolgárságot = citizenship, as an object

The ending -ot is the accusative marker.

The article is az because the next word begins with a vowel sound:

• a before most consonants
• az before vowels

So:

• az állampolgárságot = the citizenship

Those are definite-conjugation endings.

Hungarian has two main present-tense conjugation patterns:

• indefinite conjugation
• definite conjugation

You use the definite conjugation when the verb has a definite object.

Here the objects are definite:

• az állampolgárságot
• a magánútlevelet

So you get:

• igazoljuk = we verify it / let’s verify it
• tudják = they can do it

Compare:

• igazolunk egy adatot would sound different because the object is not definite in the same way
• ki tudnak állítani egy útlevelet = they can issue a passport

Különben means otherwise, or else, or sometimes if not depending on context.

In this sentence it connects the two clauses like this:

• first do X
• otherwise Y cannot happen

So the logic is:

• Igazoljuk az állampolgárságot
• különben nem tudják kiállítani a magánútlevelet

= verify the citizenship, otherwise they cannot issue the private passport.

It is a very common linking word in warnings, instructions, and conditions.

Because Hungarian often uses tud + infinitive to express can / be able to.

So:

• tudják = they can
• kiállítani = to issue

Together:

• tudják kiállítani = they can issue
• nem tudják kiállítani = they cannot issue

This is very similar to English structure:

• they can issue
• they cannot issue

Kiállítani means to issue, to produce officially, or to draw up in administrative contexts.

It is built from:

• ki- = a verbal prefix
• állít = a verb stem that in compounds can have specialized meanings

In official language, kiállít commonly means:

• issue a passport
• issue a certificate
• issue an invoice

So:

• útlevelet kiállítani = to issue a passport

A useful thing to know: Hungarian verbal prefixes can move around.

For example:

• kiállítani = to issue
• kiállítják = they issue it
• nem állítják ki = they do not issue it

With the infinitive, the prefix stays attached: kiállítani.

Because Hungarian usually leaves subject pronouns out when the verb ending already shows the subject.

Here:

• tudják already means they can or you all can depending on context
• in this sentence, it clearly means they can

So Hungarian does not need a separate pronoun like ők.

If you added ők, it would usually give extra emphasis:

• ők nem tudják kiállítani... = they cannot issue it

Without ők, it is the normal neutral way to say it.

It is an indefinite / understood they.

In administrative or official contexts, Hungarian often uses 3rd person plural when the exact people are not important, but the institution or authorities are understood.

So nem tudják kiállítani can mean something like:

• the authorities cannot issue it
• the office cannot issue it
• they cannot issue it

English sometimes does the same with they, and sometimes prefers a passive:

• it cannot be issued

Hungarian often prefers this active-looking form instead of a passive.

It is the accusative form of magánútlevél.

Breakdown:

• magán = private
• útlevél = passport
• magánútlevél = private passport
• magánútlevelet = private passport, as the object

The ending -et is the accusative marker here.

This is a compound noun, which Hungarian uses very often. In context, magánútlevél contrasts with more official or special passport types, such as diplomatic or service passports.

Because magánútlevelet begins with a consonant sound.

Hungarian articles work like this:

• a before consonants
• az before vowels

So:

• a magánútlevelet
• az állampolgárságot

This choice is based on pronunciation, not on grammatical gender, since Hungarian has no grammatical gender in articles.

The word order is quite natural.

• Igazoljuk az állampolgárságot presents the first action.
• különben introduces the consequence.
• nem tudják kiállítani a magánútlevelet gives the result if that action is not done.

Hungarian word order is flexible, but it often reflects focus and emphasis rather than just grammar.

This version sounds fairly neutral. If you wanted to emphasize a different part, the order could change. For example:

• A magánútlevelet nem tudják kiállítani...
  This puts more focus on the private passport.

So yes, the sentence is grammatical in several orders, but the given one is a normal, natural choice.