Q:

omitting the word "there"

She is used to there being no one else around.
She is used to being no one else around.

Would anyone please elaborate which one would be correct and why?

A:

She is used to there being no one else around.
*She is used to being no one else around.

Both of these sentences involve the verb used plus a subordinate clause. The second sentence is probably a grammatical mistake. Let's look at how the sentences work. Here is a similar sentence:

Maria is used to Tom being in Paris.

In this sentence, Maria is accustomed to the fact that 'Tom is in Paris'. Notice that there is a non-finite subordinate clause here.

Maria is used to [ Tom being in Paris ].

In the subordinate clause, the verb is being. This is the  --ing form of the verb BE. It is non-finite because it does not have any tense. It is not a present or past form of the verb. The Subject of this verb is Tom. Now look at the sentence below:

Maria is used to [ being in Paris ].

Here the subordinate clause also has the --ing for of the verb BE.  But there is no subject for this verb. However, we understand that the subject of the verb BE is Maria. This is because, if the Subject of the verb used and the Subject of the subordinate clause is the same person or thing, we do not need to repeat it. We can understand the sentence like this:

Mariai is used to [ ____i being in Paris ]

We can see that there is a gap in this sentence, where we expect the subject of the phrase "being in Paris" to be. The identity of this gap is controlled by the subject of the verb used.
In other words we interpret the sentence like this:
Maria is used to [ Maria being in Paris ].
This just means that Maria is used to the fact that Maria is in Paris.
The Original Poster's Question
Let's look at example 1:

She is used to there being no one else around.

Here the phrase be around just means to be close, or to be nearby. So if I live with my brother, I can say that "my brother is often around". This just means that he is often in the house with me. He is often doing things in the house. In sentence (1), we can see the same structure as in the examples above. We have the verb used and a subordinate clause:

She is used to  [there being no one else around ].

This means that she is used to the fact that There is nobody else around. In the example we see the --ing form of the verb BE. The subject of this verb is the dummy pronoun there. It is the same word we see in existential sentences like there is a new new president. The sentence is perfectly grammatical. "Nobody else around" means that she is the only person around. There is nobody apart from her in that place. So the sentence means that sometimes there is nobody around. But she is used to this.
Now let's look at the Original Poster's second example:

She is used to being no one else around.

Now this sentence has a subordinate clause. This subordinate clause does not have a subject:

She is used to  [ being no one else around ].

Again we see the --ing form of BE. It has a complement "nobody else around". Because this verb does not have a Subject, we understand the subordinate clause like this:

Shei is used to [ _____i being nobody else around ]

It is difficult to understand this clause. To make it clear let's give her a name:

Mariai is used to [ _____i being nobody else around ]

The subject gap in the subordinate clause is controlled by the subject of used. It is controlled by Maria. So we understand the sentence like this:

Maria is used to [ Maria being nobody else around ]

Hmm, this is not a good result. This sentence means that Maria is used to the fact that Maria is nobody else around. What does this mean? I don't know! It could mean that Maria is not the same person as anybody else who is around. But this is a strange meaning. I don't think this is what the Original Poster wanted to say.
This second sentence is probably a mistake. The problem is that the --ing verb doesn't have a new subject  - but, the writer does not want to say that Maria is used to Maria doing something. So this is a problem.
Hope this is helpful!
EDIT The OP has requested some resources for this grammar point. I have not been able to find any. I have written some materials that might help for practising this kind of question. You can find them in a separate post, PART 2 further below!

A:

She is used to [there being no one else around]. (gerund-participial clause in brackets)

I'd say this sentence is correct. It uses the so-called "existential there". The meaning is

She is used to the situation there there is no one else around her.

The pronoun her is missing, so we might call the gerund-participial clause a "hollow clause".
An example of a similar construction from Google Books:

True, fishers had long been accustomed to there being different closed seasons or areas for different gear types. (source)
Teachers are not accustomed to there being observers in their classrooms and are even less accustomed to using data to reflect on their own classroom practices. (source)

The second sentence:

She is used to being no one else around.

looks strange. I thought it could be accepted as an example of "poetic language" or "Yoda speech", with the meaning

She is used to being around no one else. (she is used to being alone)

But a native speaker says it's incorrect after all (see comments).
Had it been a correct sentence, it would've put more stress on the woman's own willingness to "be apart from other people", while your first example perhaps stresses more the unaccessibility of people in "there", in the location or locations she frequents.

