Viðskiptavinirnir voru að bíða í röð þegar verslunin opnaði.

-nir is the definite article suffix for masculine plural nominative nouns.
So viðskiptavinir = customers (indefinite), while viðskiptavinirnir = the customers.

Viðskiptavinur is a common masculine noun whose nominative plural typically ends in -ir.
So the base forms here are:

• singular: viðskiptavinur
• plural: viðskiptavinir
• definite plural: viðskiptavinirnir

Because the subject Viðskiptavinirnir is plural.
The past tense of vera (to be) agrees in number:

• var = was (singular)
• voru = were (plural)

This is a very common Icelandic way to express an ongoing action in the past (similar to English were waiting):

• vera (conjugated) + að
  • infinitive
    So:
• voru að bíða = were in the process of waiting / were waiting

It often emphasizes the action as in progress at a certain time.

Biðu is the simple past of bíða (wait). It can still mean waited / were waiting, but it’s usually less explicitly “in progress” than voru að bíða.
Many speakers use biðu for a straightforward narrative past, and voru að bíða when they want to highlight the ongoing nature of the waiting at that moment.

The preposition í can take:

• accusative for movement/direction (into),
• dative for location/state (in/inside).

Here the meaning is “they were waiting in a line” (a state), so you’d expect the dative.
But röð is one of those nouns where nominative/accusative/dative singular look the same: röð. So you can’t see the case change in the form.

Because the idiomatic expression is bíða í röð = wait in a line / queue.
Even if many people are involved, it’s typically thought of as one queue, so singular is normal.

In Icelandic, you usually don’t put a comma just because a subordinate clause follows (unlike in some other languages).
So ... í röð þegar verslunin opnaði is standard punctuation.

That clause is introduced by þegar (when), which makes it a subordinate clause.
In subordinate clauses, Icelandic often places the verb later in the clause (commonly after the subject), so:

• þegar verslunin opnaði = when the store opened
  This is typical subordinate-clause word order.