Hon håller på att vänja sig vid att prata svenska med sina grannar.

Håller på att + infinitive is a common way to show that something is ongoing, similar to is in the process of or sometimes is busy in English.

So:

Hon håller på att vänja sig ...
= she is in the process of getting used to ...

If you say:

Hon vänjer sig vid att prata svenska med sina grannar,
that is also correct and natural. It just sounds a little less explicitly focused on the ongoing process right now.

Because vänja sig is a reflexive verb. It means to get used to.

• vänja sig vid något = to get used to something
• vänja någon vid något = to accustom someone to something

So:

• Hon vänjer sig vid kylan = She is getting used to the cold.
• Hon vänjer barnet vid kylan = She is accustoming the child to the cold.

In your sentence, the subject herself is the one becoming accustomed, so Swedish uses sig.

Sig is the reflexive pronoun for third person. It refers back to the subject.

In this sentence, sig refers back to hon.

Compare:

• Jag vänjer mig = I am getting used to it
• Du vänjer dig = You are getting used to it
• Hon vänjer sig = She is getting used to it
• Vi vänjer oss = We are getting used to it

So sig is not extra; it is part of the verb pattern vänja sig vid.

Both can be translated as get/become used to, but they focus on slightly different things:

• vänja sig vid = the process of getting used to something
• bli van vid = becoming used to something, often closer to the result or change of state

Examples:

• Hon håller på att vänja sig vid att prata svenska
  = She is in the process of getting used to speaking Swedish.

• Hon blir van vid att prata svenska
  = She is becoming used to speaking Swedish.

In your sentence, vänja sig works especially well because the whole idea is about an ongoing adjustment.

Because they belong to two different constructions.

1. håller på att vänja sig
   Here, att is part of hålla på att + infinitive.

2. vänja sig vid att prata svenska
   Here, att introduces the infinitive phrase prata svenska after vid.

So the sentence is built like this:

• Hon håller på att
  • vänja sig
• vänja sig vid
  • att prata svenska med sina grannar

It may look repetitive, but it is completely normal Swedish.

Yes. After vänja sig vid, Swedish can use either:

• a noun
  vänja sig vid kylan = get used to the cold

or

• att + infinitive for an action
  vänja sig vid att prata svenska = get used to speaking Swedish

This is a very common pattern in Swedish. English often uses -ing after get used to, but Swedish normally uses att + infinitive here.

So:

• English: get used to speaking Swedish
• Swedish: vänja sig vid att prata svenska

Because after att, Swedish uses the infinitive form.

• att prata = to speak
• pratar = speaks / is speaking

So:

• att prata svenska = to speak Swedish
• not att pratar svenska

This is similar to English to speak, not to speaks.

In Swedish, names of languages are normally written with a lowercase letter unless they start the sentence.

So:

• svenska
• engelska
• spanska

not Svenska, Engelska, Spanska in the middle of a sentence.

Also, after verbs like prata, tala, lära sig, and förstå, the language name usually appears without an article:

• prata svenska
• lära sig engelska
• förstå tyska

So prata svenska is the normal form.

Because sina is a reflexive possessive adjective. It is used when the possessor is the same as the subject of the clause.

Here, the subject is hon, and the neighbors are her own neighbors, so Swedish uses sina.

• Hon pratar med sina grannar = She talks with her own neighbors.

If you say:

• Hon pratar med hennes grannar

that usually means She is talking with her neighbors, where her refers to some other female person, not the subject.

So sina points back to the subject; hennes does not.

After a possessive like min, din, sin, hennes, våra, etc., Swedish normally uses the noun in its indefinite form.

So you say:

• mina grannar
• sina grannar
• hennes grannar

not:

• mina grannarna
• sina grannarna

Even though the meaning is definite in English, Swedish does not usually add the definite ending when there is already a possessive.

In normal modern Swedish, the reflexive pronoun usually comes directly after the verb:

• vänja sig vid
• klä sig
• sätta sig
• bestämma sig

So:

• att vänja sig vid att prata svenska

is correct.

You would not normally move sig in front of vänja or place it after vid. The fixed pattern is:

verb + reflexive pronoun + preposition

Here: vänja + sig + vid.