In ufficio compilo un modulo per avere la nuova tessera della biblioteca.

Compilo is the first-person singular present tense of compilare.

So:

• compilare = to fill out / to complete
• compilo = I fill out / I am filling out

The sentence does not need io because Italian often leaves the subject pronoun out when the verb ending already shows who is doing the action.

Italian is a pro-drop language, which means subject pronouns are often omitted.

In English, you must say I fill out.
In Italian, compilo already tells you the subject is I, so io is usually unnecessary.

You would add io only for emphasis or contrast, for example:

• Io compilo il modulo, tu fai la fila.
  I fill out the form, you stand in line.

This is a very important false-friend point.

In Italian, compilare un modulo means to fill out a form or to complete a form.

It does not usually mean English compile in the sense of collecting information into a list or document.

So:

• compilare un modulo = to fill out a form
• English compile data is usually something else in Italian, depending on context

In ufficio means in the office or more naturally at the office / at work, depending on context.

Putting it at the beginning sets the scene first:

• In ufficio compilo... = At the office, I fill out...

Italian word order is flexible, so this could also be:

• Compilo un modulo in ufficio...

But starting with In ufficio gives a natural context-first feel.

This is a common idiomatic pattern.

In ufficio often means at work / at the office in a general sense, so Italian frequently leaves out the article.

Compare:

• in ufficio = at the office / at work, generally
• nell’ufficio = in the office, usually a specific office or room

So here, in ufficio sounds natural because the sentence is talking about the situation in a general way, not pointing to one very specific office room.

Modulo is a masculine singular noun, so the indefinite article is un:

• un modulo = a form

If it were feminine, the article would be different.

This is just the normal indefinite article pattern:

• un
  • masculine singular noun
• una
  • feminine singular noun

Per + infinitive is a very common Italian structure used to express purpose.

So:

• per avere la nuova tessera = to get the new library card / in order to have the new library card

It tells you why the speaker fills out the form.

This is similar to English to... or in order to...

Examples:

• Studio per imparare. = I study to learn.
• Lavoro per guadagnare soldi. = I work to earn money.

After the preposition per, Italian normally uses the infinitive when the subject is the same as the main verb.

Here, the same person is doing both actions:

• compilo = I fill out
• avere = to get / to have

So the structure is:

• compilo un modulo per avere...
• literally: I fill out a form to get...

If you used ho, that would be a conjugated verb and would not fit this structure.

Both are possible, but they can sound slightly different.

• la nuova tessera is the most neutral and natural way to say the new card
• la tessera nuova can sound more contrastive, like the new card (not the old one)

In many cases, common descriptive adjectives such as nuovo, vecchio, bel, grande, etc. can come before the noun, especially in everyday speech.

So here la nuova tessera is the most natural choice.

Also notice agreement:

• la tessera is feminine singular
• so the adjective is nuova

Tessera usually means a card used for membership, identification, access, or registration.

In this sentence, tessera della biblioteca means library card.

It is not exactly the same as every use of English card. For example:

• tessera = membership/access card, library card, club card
• carta can mean paper, map, or sometimes card depending on context
• carta di credito = credit card

So tessera is the right word here because a library card is a membership/access-type card.

Della is the combination of:

• di
  • la = della

Here it expresses a relationship:

• la tessera della biblioteca = the library’s card / the library card

Italian often uses di + article + noun where English uses a noun directly before another noun.

So English says:

• library card

But Italian says:

• tessera della biblioteca

This is very normal.

Because Italian usually does not build noun + noun combinations as freely as English does.

English often uses one noun to describe another:

• library card
• office worker
• train station

Italian very often prefers a structure with di:

• tessera della biblioteca
• stazione del treno is not the usual phrase, but the general idea is the same: Italian often needs a linking preposition or a different construction

So tessera della biblioteca is the natural Italian way.

This is a very common mistake for English speakers.

• biblioteca = library
• libreria = bookstore / bookshop
  and sometimes bookcase, depending on context

So in this sentence:

• biblioteca is correct because it means library

Do not use libreria if you mean a place where you borrow books.

Yes. Italian word order is fairly flexible.

For example, these are all possible:

• In ufficio compilo un modulo per avere la nuova tessera della biblioteca.
• Compilo un modulo in ufficio per avere la nuova tessera della biblioteca.
• Per avere la nuova tessera della biblioteca, compilo un modulo in ufficio.

They all mean roughly the same thing, but the emphasis changes:

• In ufficio... puts the location first
• Per avere... puts the purpose first
• starting with Compilo... focuses first on the action

The original sentence sounds natural and straightforward.