Cerco un’azienda in città.

Cerco is the first-person singular present indicative of cercare.

So:

• io cerco = I look for / I am looking for
• tu cerchi = you look for
• lui/lei cerca = he/she looks for

In Italian, the simple present often covers both:

• a general meaning: I look for
• a current action: I am looking for

So cerco is perfectly natural here.

Because cercare normally takes a direct object in Italian.

English says:

• I’m looking for a company

Italian says:

• Cerco un’azienda

So you do not add per here.
A sentence like cerco per un’azienda would be wrong in this meaning.

This is a very common difference between English and Italian:

• to look for something → cercare qualcosa

Because azienda is a feminine singular noun that begins with a vowel, so una becomes un’ before it.

So:

• una casa
• una città
• un’azienda
• un’amica

That apostrophe shows elision: the final vowel of una drops before another vowel.

Important:

• un’ with an apostrophe = feminine
• un without an apostrophe can be masculine, as in un amico

Its article is feminine:

• l’azienda
• un’azienda

Its plural is also feminine:

• le aziende

Italian nouns do not always have obvious endings, so sometimes you simply have to learn the gender with the word. A good habit is to learn nouns together with their article:

• l’azienda
• la città
• il libro

Azienda usually means company, business, firm, or enterprise.

It is often used for a business organization rather than a physical store. So depending on context, it may sound more like:

• a company
• a business
• an enterprise

If you specifically mean a shop/store, Italian often uses negozio instead.

Related words:

• azienda = company/business
• impresa = enterprise/business
• ditta = firm/company
• negozio = shop/store

In città is a very common idiomatic expression meaning in town or in the city in a general sense.

Italian often leaves out the article in set location expressions like this:

• in città
• in centro
• in ufficio
• a casa

Nella città is possible, but it usually sounds more specific, as if you mean in the city itself, often in contrast with somewhere else.

So:

• Cerco un’azienda in città = natural, general
• Cerco un’azienda nella città = more marked, more specific

Because città is stressed on the last syllable, and in Italian final stress is usually marked with a written accent.

So:

• cit-TÀ

This accent is part of the spelling.

A useful extra point: città is one of those nouns that does not change form in the plural.

So:

• la città = the city
• le città = the cities

The article changes, but the noun stays the same.

A rough pronunciation guide is:

CHER-ko oo-nah-TSYEN-dah een cheet-TAH

More carefully:

• cerco = CHER-ko
  
  • first c before e sounds like ch in chair
  • second c before o sounds like k
• un’azienda = roughly oo-nah-TSYEN-dah
• in città = een cheet-TAH

A more Italian-like pronunciation would be:

• cerco /ˈtʃɛrko/
• azienda /adˈdzjɛnda/
• città /tʃitˈta/

Yes. That would also be correct.

Difference:

• Cerco un’azienda in città = simple, neutral, very common
• Sto cercando un’azienda in città = emphasizes that the search is happening right now

Italian uses the simple present more often than English does, so cerco is often enough even when English would say I’m looking for.

By default, most people will understand in città as describing the company: you are looking for a company located in the city.

But technically, context can make it a little flexible.

Possible readings:

• you are looking for a company in the city
• less likely, you are in the city and looking for a company

If you want to make the location of the company clearer, you could say:

• Cerco un’azienda che sia in città.

If you want to emphasize where the searching happens, you could say:

• Cerco in città un’azienda.

Yes. Cerco un’azienda in città is normal, neutral Italian word order.

Basic structure:

• Cerco = verb
• un’azienda = direct object
• in città = location phrase

Italian word order is often flexible, but this version is the most straightforward and natural in ordinary speech. Other orders are possible, but they usually add emphasis or sound more marked.