Az alvázszám és a rendszám a forgalmi engedélyben van.

A and az both mean the. Hungarian uses:

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

So:

• az alvázszám because alvázszám starts with a
• a rendszám because rendszám starts with r
• a forgalmi engedélyben because the whole noun phrase starts with f

This is similar to a/an in English, but it applies to the in Hungarian.

Hungarian very often repeats the definite article with each definite noun in a coordinated phrase.

So az alvázszám és a rendszám is the natural full form:
the chassis number and the license plate number

If you left out the second article, it would usually sound less natural here. Repeating the article makes it clear that both nouns are definite and separate items.

Alvázszám is a compound noun:

• alváz = chassis / frame
• szám = number

So literally it is chassis number.

Hungarian forms a lot of compounds, and they are usually written as one word, much more often than in English. The same is true for rendszám:

• rend / rendszer-related historical root
• szám = number

In modern usage, rendszám means license plate number or sometimes loosely license plate.

The ending -ban / -ben means in.

So:

• engedély = permit / certificate
• engedélyben = in the permit / certificate

Hungarian usually uses suffixes where English uses prepositions.

The choice between -ban and -ben depends on vowel harmony:

• back vowels usually take -ban
• front vowels usually take -ben

Engedély has front vowels, so it becomes engedélyben.

Yes, forgalmi is functioning like an adjective here, modifying engedély.

• forgalom = traffic / circulation / use
• forgalmi = traffic-related / registration-related
• engedély = permit, certificate

Together, forgalmi engedély is the standard term for a vehicle registration certificate or registration document.

In everyday Hungarian, people very often shorten this and simply say a forgalmi.

That is a very common question.

Hungarian often omits van in 3rd person singular/plural present tense when the sentence is simply describing or identifying something:

• A kocsi piros. = The car is red.
• Ez a könyv új. = This book is new.

But van is normally kept in sentences of location or existence:

• A könyv az asztalon van. = The book is on the table.
• A kulcs a táskában van. = The key is in the bag.

Your sentence is a location sentence, so van is needed.

This is one of those places where Hungarian does not always match English expectations.

With two inanimate things joined by és, Hungarian often uses singular van in a location/existence sentence when the items are treated as a kind of set or grouped information.

So Az alvázszám és a rendszám a forgalmi engedélyben van is natural.

Using vannak is not impossible in every context, but van is very common and sounds normal here.

A useful contrast:

• Péter és Anna otthon vannak. = Peter and Anna are at home.
  With people, plural vannak is expected.

• A kés és a villa az asztalon van. = The knife and fork are on the table.
  With objects, singular van is often used.

Yes, you could say that, but the emphasis changes.

Hungarian word order is much more flexible than English word order, and it often reflects topic and focus rather than strict grammar.

Az alvázszám és a rendszám a forgalmi engedélyben van.
This is a fairly neutral way to say that those items are in the registration document.

A forgalmi engedélyben van az alvázszám és a rendszám.
This puts more emphasis on where they are: It is in the registration document that the chassis number and plate number are.

Both are grammatical; the difference is mainly information structure and emphasis.

Because forgalmi engedély is one noun phrase, and the case ending goes on the last word of the phrase.

So:

• a forgalmi engedély = the registration certificate
• a forgalmi engedélyben = in the registration certificate

This is very normal in Hungarian. The modifier stays unchanged, and the main noun gets the ending.

Yes. In everyday Hungarian, a forgalmi very often means the vehicle registration document by itself.

So in conversation, you may hear things like:

• Hol a forgalmi? = Where is the registration document?
• A forgalmiban van. = It’s in the registration document / It’s in the registration papers.

This is a common shorthand, but the full form is forgalmi engedély.