Potrzebuję zarówno formularza odstąpienia, jak i dowodu zakupu.

It is a fixed pair meaning both ... and ....

So:

• zarówno formularza odstąpienia, jak i dowodu zakupu = both the withdrawal form and the proof of purchase

It adds emphasis. Without that emphasis, you could also simply say:

• Potrzebuję formularza odstąpienia i dowodu zakupu.

That still means I need the withdrawal form and the proof of purchase, but zarówno ..., jak i ... makes the both idea more explicit.

Potrzebuję is the 1st person singular form of potrzebować, which means to need.

So:

• potrzebować = to need
• potrzebuję = I need
• potrzebujesz = you need
• potrzebuje = he/she/it needs

In this sentence, the speaker is saying I need..., so potrzebuję is the correct form.

Because potrzebować normally requires the genitive case.

The basic dictionary forms are:

• formularz = form
• dowód = proof, document

But after potrzebuję, they change to genitive:

• potrzebuję formularza
• potrzebuję dowodu

This is a very common pattern in Polish:

• Potrzebuję czasu = I need time
• Potrzebuję pomocy = I need help
• Potrzebuję dokumentu = I need a document

So the short answer is: potrzebować + genitive.

Because they depend on the nouns before them.

• formularz odstąpienia = withdrawal form
• dowód zakupu = proof of purchase

Here, odstąpienia depends on formularz, and zakupu depends on dowód.

This is another common Polish pattern: one noun is followed by another noun in the genitive, similar to English form of withdrawal or proof of purchase.

So there are really two things happening:

1. formularza and dowodu are genitive because of potrzebuję
2. odstąpienia and zakupu are genitive because they depend on formularz and dowód

In everyday Polish, odstąpienie in this context is usually a legal or formal withdrawal from a purchase/contract.

So formularz odstąpienia is not just any random form. It usually means something like:

• withdrawal form
• cancellation form
• in online shopping contexts, often the form used to withdraw from a distance purchase

If you are dealing with returns, Polish may also use zwrot in many situations, but odstąpienie sounds more formal/legal.

Literally, it means proof of purchase.

• dowód = proof, evidence, document
• zakup = purchase

In practice, dowód zakupu can mean things like:

• a receipt
• an invoice
• another document showing that you bought the item

So even though dowód can mean several things in Polish, here it clearly means proof/evidence.

Because Polish nouns do not all form the genitive singular in the same way.

Both nouns are masculine, but their genitive endings are different:

• formularz → formularza
• dowód → dowodu

This is normal in Polish. Masculine nouns often take either -a or -u in the genitive singular, depending on the word.

Unfortunately, there is not always a simple rule that works for every noun, so learners often have to learn these forms word by word.

Because Polish has no articles.

English needs words like:

• a
• an
• the

Polish does not. So formularza can mean:

• a form
• the form

and dowodu zakupu can mean:

• proof of purchase
• the proof of purchase

The exact meaning is understood from context.

Yes.

You can say:

• Potrzebuję zarówno formularza odstąpienia, jak i dowodu zakupu.

or simply:

• Potrzebuję formularza odstąpienia i dowodu zakupu.

The second version is simpler and very natural. The first version adds emphasis to the idea of both items being required.

Polish word order is fairly flexible, but this sentence is already in a very natural order.

The pattern zarówno ..., jak i ... is usually kept together like this:

• zarówno X, jak i Y

So the most natural version is:

• Potrzebuję zarówno formularza odstąpienia, jak i dowodu zakupu.

If you move things around too much, the sentence may sound awkward or overly marked. For a learner, it is best to keep this structure as it is.