Ég veit ekki hvaða leigubíl við eigum að panta þegar það er orðið svona seint.

Leigubíl is in the accusative singular because it is the direct object of panta (to order / book).

The base noun is leigubíll (taxi). Its accusative singular form is leigubíl.

Hvaða means which / what kind of here, and it does not change form for gender, number, or case, so you still say hvaða leigubíl.

Because this is an embedded question, not a direct question.

• Direct question: Hvaða leigubíl eigum við að panta?
  = Which taxi should we order?

• Embedded question: Ég veit ekki hvaða leigubíl við eigum að panta.
  = I don't know which taxi we should order.

In Icelandic, embedded questions usually keep statement word order, so the subject við comes before the finite verb phrase in the clause.

Eiga að + infinitive is a very common Icelandic structure meaning:

• should
• ought to
• are supposed to
• sometimes must, depending on context

So við eigum að panta means we should order / we are supposed to order.

Even though eiga by itself usually means to own / have, in this structure it has a different function.

After eiga in the sense of should / ought to, Icelandic normally uses að + infinitive.

So:

• eiga að fara = should go
• eiga að panta = should order

Here, panta stays in the infinitive because it depends on eigum að.

Yes. Panta means order, book, or reserve, depending on context.

With a taxi, English often says:

• call a taxi
• book a taxi
• order a taxi

Icelandic commonly uses panta in this situation, so panta leigubíl is a normal way to say book/order a taxi.

Er orðið is the present perfect of verða (to become), so it literally means has become.

So:

• það er svona seint = it is this late / it is so late
• það er orðið svona seint = it has gotten this late / it has become this late

Using orðið emphasizes that time has reached this late point. In English, we might or might not express that explicitly.

Orðið is the past participle of verða (to become).

In það er orðið svona seint, the structure is:

• er = is / has
• orðið = become

Together: has become / has gotten

This is a very common Icelandic pattern:

• Það er orðið kalt. = It has gotten cold.
• Hann er orðinn þreyttur. = He has become tired.

This það works like English it in expressions about time, weather, and general conditions.

So just as English says:

• it is late
• it is cold

Icelandic says:

• það er seint
• það er kalt

It does not refer to a specific thing; it is a dummy subject.

Svona means like this, this way, or in many contexts this / so before an adjective or adverb.

In svona seint, it means this late or so late.

So:

• svona seint = this late
• svona stórt = this big
• svona mikið = this much / so much

It adds emphasis and points to the degree.

Here þegar is basically when, but in natural English the whole clause can sometimes sound like:

• when it’s gotten this late
• once it gets this late
• now that it’s this late

The exact feel depends on context. Grammatically, it introduces a time clause.

In a main clause, Icelandic ekki normally comes after the finite verb.

So:

• Ég veit ekki = I do not know
• Hann kemur ekki = He is not coming

That is the normal placement of negation in simple main clauses.

Yes, but then the wording would change.

The sentence you have is a statement:

• Ég veit ekki hvaða leigubíl við eigum að panta ...
• I don’t know which taxi we should order ...

If you want the direct question itself, you would say:

• Hvaða leigubíl eigum við að panta þegar það er orðið svona seint?
• Which taxi should we order when it’s gotten this late?

The main difference is the word order in the question clause.

Yes. Við is the subject of eigum að panta, so it is in the nominative case.

The same is true for:

• ég in Ég veit ekki
• það in það er orðið svona seint

Subjects are normally nominative in Icelandic.

Because the meaning is which taxi, not which specific taxi that we already both know about.

• hvaða leigubíl = which taxi
• hvaða leigubílinn would sound much more specific and marked, like which taxi is it, the one we mean?

In ordinary usage, hvaða + noun is usually indefinite in this kind of sentence.

It is natural everyday Icelandic.

The structure is very common:

• Ég veit ekki... = I don’t know...
• embedded question with hvaða
• eiga að + infinitive
• time clause with þegar
• condition/state with það er orðið svona seint

A native speaker could easily say this in normal conversation.