A panaszban kérem az ügyfélszolgálatot, hogy vizsgálja ki a kézbesítést.

panasz means complaint.

The ending -ban / -ben means in. So:

• panasz = complaint
• panaszban = in the complaint

The article a means the, so A panaszban = In the complaint.

This phrase sets the context for the whole sentence: it tells you that the request is being made in the complaint document/text.

Because it is the direct object of kérem.

• ügyfélszolgálat = customer service
• ügyfélszolgálatot = customer service + accusative ending

In Hungarian, the direct object usually gets -t (here it appears as -ot because of vowel harmony and pronunciation).

So:

• kérem az ügyfélszolgálatot = I ask customer service

English does not mark this with a special ending, but Hungarian does.

Hungarian verbs can have definite and indefinite conjugation.

Here the verb kér takes a definite object:

• az ügyfélszolgálatot = the customer service

Because the object is definite, Hungarian uses the definite conjugation:

• kérem = I ask it / I request it

Compare:

• kérek valamit = I ask for something / I request something
• kérem az ügyfélszolgálatot = I ask the customer service

So kérem is the correct form here.

No, not in this sentence.

kérem can sometimes mean please in polite expressions, but here it clearly means I ask / I request because it has:

• a direct object: az ügyfélszolgálatot
• a following clause: hogy vizsgálja ki...

So in this sentence, kérem is a full verb, not just a polite formula.

After verbs like ask, request, want, Hungarian very often uses hogy + a finite verb.

So:

• kérem ..., hogy ... = I ask ..., to ... / that ...

English often uses an infinitive:

• I ask customer service to investigate...

Hungarian commonly expresses this with a whole clause:

• kérem az ügyfélszolgálatot, hogy vizsgálja ki...

Literally, it is closer to:

• I ask customer service that it investigate...

That sounds strange in English, but it is normal in Hungarian.

Because this clause expresses something being requested, not just a plain statement.

After kérem ..., hogy ..., Hungarian uses the imperative/subjunctive-type form:

• vizsgálja = that he/she/it investigate

By contrast:

• vizsgál = he/she/it investigates
  This would just be a statement, not the requested action.

So vizsgálja is used because the speaker is asking customer service to do the investigation.

Both forms are related to the imperative/subjunctive, but they differ in conjugation.

Here the verb has a definite object:

• a kézbesítést = the delivery

Because of that, Hungarian uses the definite form:

• vizsgálja

Compare:

• vizsgáljon valamit = investigate something
• vizsgálja a kézbesítést = investigate the delivery

So vizsgálja matches the definite object a kézbesítést.

ki is a verbal prefix.

The idea is similar to the verb kivizsgál, which means investigate thoroughly / look into fully.

In request or command-like forms, Hungarian often places the prefix after the verb:

• kivizsgálja → vizsgálja ki

So vizsgálja ki means something like:

• investigate it fully
• look into it
• carry out an investigation of it

This is a very normal Hungarian pattern.

Because in Hungarian, verbal prefixes often move after the verb in:

• commands
• requests
• imperative/subjunctive forms

So although the dictionary form may look like kivizsgál, in a clause like this you often get:

• hogy vizsgálja ki

This is standard Hungarian syntax, not an exception.

kézbesítés means delivery, especially the act or process of delivering something to the recipient.

So it refers to the delivery process/event, not the package itself.

• kézbesítés = delivery
• kézbesítést = delivery + accusative

It has -t because it is the object of vizsgálja ki:

• investigate the delivery

Hungarian word order is flexible, and the beginning of the sentence often gives the topic or frame.

Starting with A panaszban tells us:

• As for the complaint...
• In the complaint...

So the sentence first sets the context, then gives the main action.

This is especially natural in formal written language.

Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

So:

• az ügyfélszolgálatot because ügyfélszolgálat begins with the vowel ü
• a kézbesítést because kézbesítés begins with the consonant k

This is just like choosing the smoother-sounding article form.

Yes, it is grammatically singular.

In Hungarian, ügyfélszolgálat usually refers to the customer service department/unit as one entity, even though several people may work there.

That is why the verb is singular:

• hogy vizsgálja ki

If you wanted to focus on the individual employees, you would normally phrase the sentence differently. But with ügyfélszolgálat, the singular is the natural choice.