שירות הלקוחות שלח כרטיס חלופי.

Because the subject of the sentence is שירות הלקוחות, not לקוחות by itself.

• שירות = service
• הלקוחות = the customers

Together, שירות הלקוחות means customer service or more literally the service of the customers / the customer-service department.

The head noun is שירות, and שירות is:

• singular
• masculine

So the verb agrees with שירות, which is why you get שלח = sent (masculine singular past).

It is a construct phrase in Hebrew.

• שירות = service
• הלקוחות = the customers

So literally it is service of the customers, which is the normal Hebrew way to say customer service.

This pattern is very common in Hebrew:

• בית ספר = school, literally house of book
• חדר אוכל = dining room, literally room of food
• שירות הלקוחות = customer service

In a construct phrase, the first noun often does not take ה־ even when the whole phrase is definite. That is why it is שירות הלקוחות, not השירות הלקוחות.

Because in this kind of noun chain, definiteness is usually marked on the second noun.

So:

• שירות לקוחות = customer service / customer-service
• שירות הלקוחות = the customer service

Even though שירות itself has no ה־, the whole phrase can still be definite because the second noun is definite.

This is a standard rule in Hebrew construct phrases.

Because את is used before a definite direct object, and כרטיס חלופי is indefinite.

• כרטיס חלופי = a replacement card
• הכרטיס החלופי = the replacement card

So:

• שירות הלקוחות שלח כרטיס חלופי = Customer service sent a replacement card.
• שירות הלקוחות שלח את הכרטיס החלופי = Customer service sent the replacement card.

Since the object here is not definite, את is not used.

Because in Hebrew, adjectives usually come after the noun.

So:

• כרטיס = card
• חלופי = replacement / alternative

Together:

• כרטיס חלופי = a replacement card

This is the normal Hebrew word order:

• בית גדול = a big house
• ילדה חכמה = a smart girl
• כרטיס חלופי = a replacement card

Because adjectives in Hebrew must agree with the noun in:

• gender
• number
• definiteness

Here, כרטיס is:

• masculine
• singular
• indefinite

So the adjective is also:

• masculine singular indefinite → חלופי

Compare:

• כרטיס חלופי = a replacement card
• כרטיסים חלופיים = replacement cards
• תעודה חלופית = a replacement document/certificate (feminine singular)
• הכרטיס החלופי = the replacement card

שלח is the past tense, third person masculine singular form of the verb לשלוח = to send.

So:

• לשלוח = to send
• שלח = he sent / it sent

In this sentence, it can be translated as sent, and the subject שירות הלקוחות is treated like it in English:

• Customer service sent a replacement card.

Some related forms:

• שלחה = she sent
• שלחו = they sent
• שולח = sends / is sending (masculine singular, present)
• ישלח = he will send

Not with שירות הלקוחות, because שירות is masculine.

You would use שלחה only if the subject were feminine singular. For example:

• החברה שלחה כרטיס חלופי = The company sent a replacement card.
• המחלקה שלחה כרטיס חלופי = The department sent a replacement card.

But:

• שירות הלקוחות שלח כרטיס חלופי = correct, because שירות is masculine.

In a sentence like this, it usually refers to the customer service department/team as an acting entity.

So even though the literal words are customer service, in practice it often means:

• customer service
• the customer service department
• the customer support team

Hebrew often uses such nouns as organizational subjects:

• הבנק שלח הודעה = The bank sent a notice.
• האוניברסיטה הודיעה = The university announced...
• שירות הלקוחות שלח כרטיס חלופי = Customer service sent a replacement card.

A simple pronunciation is:

Sherut ha-lakhochot shalakh kartis khalufi.

A slightly more careful breakdown:

• שירות → she-RUT
• הלקוחות → ha-la-KHO-khot
• שלח → sha-LAKH
• כרטיס → kar-TIS
• חלופי → kha-lu-FI

Notes:

• The sound written here as kh is the throaty Hebrew sound in letters like ח and sometimes כ without dagesh.
• Stress is usually near the end in these words.

You would make the direct object definite, and then you would also add את:

שירות הלקוחות שלח את הכרטיס החלופי.

Why?

• הכרטיס החלופי = the replacement card
• A definite direct object in Hebrew usually takes את

So compare:

• שירות הלקוחות שלח כרטיס חלופי = Customer service sent a replacement card.
• שירות הלקוחות שלח את הכרטיס החלופי = Customer service sent the replacement card.

Yes. The adjective חלופי can mean something like:

• replacement
• substitute
• alternative

The exact meaning depends on context.

So כרטיס חלופי can mean:

• a replacement card
• a substitute card

In the context of banks, memberships, transit cards, SIM cards, and similar things, replacement card is usually the most natural translation.