Ona nie tylko sprawdza mój formularz, ale też stawia na nim pieczątkę.

Yes. Polish often drops subject pronouns because the verb already shows the person.

• sprawdza = she/he/it checks
• stawia = she/he/it puts/places

So a very natural version is simply:

Nie tylko sprawdza mój formularz, ale też stawia na nim pieczątkę.

Adding ona gives extra emphasis or contrast, like:

• She not only checks my form, but also stamps it.
• It is she who does both things.

So ona is not required, but it is possible when you want to stress the subject.

It is the Polish equivalent of not only ..., but also ...

In this sentence:

• nie tylko sprawdza mój formularz = she not only checks my form
• ale też stawia na nim pieczątkę = but also puts a stamp on it

A few useful points:

• ale też is very common and natural.
• You may also see ale także, which means the same thing and can sound a bit more formal.
• The two parts usually match each other structurally, just like in English.

So this pattern is worth learning as a fixed expression: nie tylko X, ale też Y

They are in the 3rd person singular present tense.

• sprawdza = she checks / is checking
• stawia = she puts / is putting

Because the subject is ona (she), both verbs must be in the she/he/it form.

Also, these are imperfective verbs, which is important in Polish. Imperfective verbs are used for:

• ongoing actions
• repeated actions
• general present-time statements

That is why sprawdza and stawia fit here.

If you used perfective verbs instead, their present-looking forms would usually refer to the future, not the present.

Because formularz is the direct object of sprawdza, so it is in the accusative case.

However, formularz is a masculine inanimate noun, and in Polish, masculine inanimate nouns often have the same form in the accusative as in the nominative.

So:

• nominative: mój formularz
• accusative: mój formularz

That is why nothing seems to change.

Compare that with a masculine animate noun:

• nominative: mój pies
• accusative: mojego psa

So the lack of change in mój formularz is normal.

na nim is in the locative case.

The preposition na can take different cases depending on the meaning:

• na + accusative = movement onto something
• na + locative = location on something

Here there is no movement onto the form. The stamp is being placed on the form as a location/surface, so Polish uses na + locative:

• na formularzu = on the form
• na nim = on it

So na nim means on it, referring back to formularz.

Because after a preposition, Polish personal pronouns usually take a special prepositional form.

Here the pronoun refers to formularz, and after na you need:

• na nim = on it

You cannot say na go.

This is a general rule in Polish. Compare:

• widzę go = I see him/it
• mówię o nim = I am talking about him/it
• na nim = on it

So nim is the correct pronoun form after the preposition na.

Because pieczątkę is in the accusative singular.

The basic form is:

• nominative: pieczątka = stamp

As the direct object of stawia, it changes to the accusative:

• accusative: pieczątkę

This is very common for many feminine nouns ending in -a:

• książka → książkę
• kawa → kawę
• pieczątka → pieczątkę

So stawia pieczątkę means puts/adds a stamp.

Yes, it is a normal expression.

Literally, stawia pieczątkę is something like puts/places a stamp, but in natural English it often corresponds to:

• stamps it
• puts a stamp on it
• affixes a stamp
• applies a stamp

Other Polish verbs can also appear with pieczątka, depending on context, for example:

• przybić pieczątkę — often used for physically stamping something
• podbić — often used for validating a document with a stamp

But stawia pieczątkę is perfectly understandable and natural.

It refers to mój formularz.

So:

• mój formularz = my form
• na nim = on it

Polish often uses a pronoun like this instead of repeating the noun:

• stawia pieczątkę na formularzu
• stawia na nim pieczątkę

Both are possible. The second version avoids repetition and sounds natural.

Polish word order is fairly flexible, although some versions sound more neutral than others.

Your sentence is correct:

Ona nie tylko sprawdza mój formularz, ale też stawia na nim pieczątkę.

Other natural possibilities include:

• Nie tylko sprawdza mój formularz, ale też stawia na nim pieczątkę.
• Ona nie tylko sprawdza mój formularz, ale stawia też na nim pieczątkę.

The differences are mostly about emphasis.

A few general points:

• putting ona first emphasizes the subject
• putting też earlier or later slightly shifts emphasis
• the nie tylko ..., ale też ... structure should stay clear

So the sentence is flexible, but not completely random. Polish word order often changes to highlight different parts of the sentence.