Klientka pyta kelnerkę, czy może zapłacić kartą.

Because they have different grammatical roles in the sentence.

• Klientka is the subject, so it stays in the nominative form.
• kelnerkę is the person being asked, and the verb pytać takes the accusative case: pytać kogo? = to ask whom?

So:

• klientka = the female customer
• pyta kelnerkę = asks the waitress

The base form is kelnerka, but in the accusative singular it becomes kelnerkę.

Because pytać requires the accusative, not the dative.

You ask:

• pytać kogo? o co? = to ask someone about something

So:

• pyta kelnerkę = she asks the waitress

By contrast, kelnerce is dative, and would fit with verbs like:

• mówić kelnerce = to speak to / tell the waitress
• dać kelnerce = to give to the waitress

So this is a very common distinction:

• pytać kogoś = ask someone
• mówić komuś = tell/speak to someone

Czy introduces an indirect yes/no question.

In English, we would say:

• The customer asks the waitress whether she can pay by card.

That whether / if idea is expressed by czy in Polish.

So:

• Klientka pyta kelnerkę, czy może zapłacić kartą. = The customer asks the waitress whether she can pay by card.

Important: this czy is not the same as the English or in a choice question. It simply marks a yes/no clause.

You also use czy in direct questions:

• Czy mogę zapłacić kartą? = Can I pay by card?

Because in Polish, the subject is often omitted when it is clear from the verb form or the context.

• może = he/she/it can / may

Polish does not need to say ona every time.

Here, from context, we understand that it is the customer asking whether she can pay by card.

So Polish naturally says:

• czy może zapłacić kartą

rather than:

• czy ona może zapłacić kartą

Adding ona is possible, but it usually sounds more emphatic or contrastive.

It can correspond to both, depending on context.

• może comes from móc = to be able to / can
• In many real-life situations, English can and may overlap

So here czy może zapłacić kartą could be understood as:

• whether she can pay by card
• whether she may pay by card
• whether it is possible / allowed to pay by card

In everyday translation, can pay by card is the most natural English version.

Because zapłacić is the perfective verb, and it refers to completing a single payment.

Polish often uses verb pairs:

• płacić = imperfective, paying / to pay in general / repeatedly
• zapłacić = perfective, to pay once, to make the payment, to finish paying

In this sentence, the customer is asking about one complete action:

• Can I pay now, by card?

So zapłacić is the natural choice.

Compare:

• Lubię płacić kartą. = I like paying by card.
  (general habit)
• Mogę zapłacić kartą? = Can I pay by card?
  (one completed payment)

Because kartą is the instrumental case, used here to express the means or instrument by which something is done.

So:

• zapłacić kartą = pay by card / with a card

The basic form is:

• karta = card

Instrumental singular:

• kartą

This is very common in Polish when talking about the means of doing something:

• jechać autobusem = go by bus
• pisać długopisem = write with a pen
• zapłacić gotówką = pay in cash
• zapłacić kartą = pay by card

So English uses by or with, while Polish often uses the instrumental case.

By itself, karta can mean different kinds of card depending on context, but in this sentence kartą clearly means a payment card.

In everyday Polish:

• płacić kartą almost always means to pay by card
• usually a debit or credit card

If needed, Polish can be more specific:

• kartą kredytową = by credit card
• kartą debetową = by debit card

But in a restaurant/shop context, kartą is completely natural and usually unambiguous.

Yes. That is perfectly natural.

• Klientka pyta kelnerkę, czy może zapłacić kartą.
• Klientka pyta, czy może zapłacić kartą.

The first version explicitly says whom she is asking: the waitress.
The second version leaves that unsaid because it may already be obvious from context.

Polish often omits information that is already understood.

No, Polish word order is fairly flexible, though some orders sound more neutral than others.

The most neutral version here is:

• Klientka pyta kelnerkę, czy może zapłacić kartą.

But other orders are possible, for example:

• Klientka pyta, czy może zapłacić kartą, kelnerkę.
  This sounds awkward, so not all rearrangements are equally good.
• Czy klientka może zapłacić kartą?
  This changes the sentence into a direct question.
• Kartą klientka może zapłacić?
  Possible only in a marked context, with unusual emphasis.

So yes, Polish word order is flexible, but learners should stick to the neutral order until they feel comfortable with emphasis and information structure.

Because czy może zapłacić kartą is a subordinate clause, and Polish normally separates such clauses with a comma.

So:

• Klientka pyta kelnerkę, czy może zapłacić kartą.

This is standard Polish punctuation.

The comma marks the boundary between:

• the main clause: Klientka pyta kelnerkę
• the subordinate clause: czy może zapłacić kartą

English punctuation may be lighter in some cases, but in Polish this comma is expected.

You would change the gendered nouns and some endings:

• Klient pyta kelnera, czy może zapłacić kartą.

Changes:

• klientka → klient = female customer → male customer
• kelnerkę → kelnera if the waiter is male

If the customer is male but the waitress is female:

• Klient pyta kelnerkę, czy może zapłacić kartą.

The verb pyta stays the same in the present tense.

Yes, it can absolutely mean customer.

In Polish:

• klient = male client/customer
• klientka = female client/customer

In many everyday situations, klient / klientka is best translated as customer, not just client in the narrow business sense.

So in a restaurant context, klientka is naturally female customer.

A rough pronunciation guide is:

• Klientka ≈ klyent-ka
• pyta ≈ PIH-ta
• kelnerkę ≈ kel-NER-keh
• czy ≈ chi (but with a harder Polish cz sound)
• może ≈ MO-zhe
• zapłacić ≈ za-PWA-chich
• kartą ≈ KAR-ton (with nasal ą)

A few useful sound notes:

• cz is like the ch in chop, but retroflex/harder
• ż sounds like the s in measure
• ł sounds like English w
• ą is a nasal vowel; at the end of a word it often sounds somewhat like on/om

So zapłacić kartą can sound roughly like:

• za-PWA-chich KAR-tom

That is only approximate, but it helps as a first step.