Ég stend við bensíndæluna á meðan hún borgar inni.

Standa is the infinitive, meaning to stand.
In the sentence, the verb has to match ég = I, so you need the 1st person singular present form:

• ég stend = I stand / I am standing
• þú stendur = you stand
• hún stendur = she stands

This verb is irregular, so the stem changes from stand- to stend- in ég stend.

Icelandic usually does not have a separate progressive form like English am standing or is paying.
The ordinary present tense often covers both meanings:

• ég stend can mean I stand or I am standing
• hún borgar can mean she pays or she is paying

So the ongoing sense comes from context, not from a special verb form.

Here við means by, at, or next to.
So við bensíndæluna means by the gas pump or at the gas pump.

It is a very natural preposition for being positioned beside something.

Bensíndæluna is the accusative singular definite form of bensíndæla, which means gas pump.

You can break it down like this:

• bensíndæla = gas pump
• bensíndæluna = the gas pump

The ending changes for two reasons:

1. við takes the accusative case
2. the noun is definite, so Icelandic adds the as a suffix at the end of the noun

So this is not a separate word for the. Icelandic attaches it to the noun.

Yes. It is made from:

• bensín = petrol, gas
• dæla = pump

So bensíndæla literally means petrol pump or gas pump.
Compound nouns are extremely common in Icelandic.

Á meðan is a fixed expression meaning while.
It introduces a subordinate clause that happens at the same time as the main clause:

• Ég stend við bensíndæluna = main clause
• á meðan hún borgar inni = while she pays inside

You should learn á meðan as a set phrase here.

Because after a subordinating expression like á meðan, Icelandic normally uses subordinate clause word order:

• á meðan + subject + verb + other elements

So:

• á meðan hún borgar inni = correct

In main clauses, Icelandic often follows a verb-second pattern, but subordinate clauses do not behave the same way. That is why hún comes before borgar here.

Inni means inside or indoors.
In this sentence it means she is paying inside the building, probably inside the gas station shop or cashier area.

It is an adverb, so it can stand on its own without naming the place again. Icelandic often does this when the location is obvious from context.

Because inni describes location, not movement.

• inni = inside, being inside
• inn often relates to motion inward

Here she is located inside while paying, so inni is the natural choice.

Because Icelandic, like English, can leave the object unstated when it is obvious from context.

At a gas station, hún borgar inni naturally suggests that she is going inside to pay for the fuel or the purchase. You do not need to spell it out.

So borga can work this way when the listener already understands what is being paid for.

Normally no. Icelandic usually requires an expressed subject in a finite clause.

So you say:

• á meðan hún borgar inni

not just:

• á meðan borgar inni

The verb borgar tells you only that the subject is 3rd person singular. It does not by itself make the subject clear enough, so hún is needed.

In modern Icelandic, it is very normal not to use a comma before a subordinate clause like this when it comes at the end of the sentence.

So this is perfectly natural:

• Ég stend við bensíndæluna á meðan hún borgar inni.

You may sometimes see commas in older writing or in different punctuation styles, but no comma here is standard and natural.