코로나 바이러스가 1.5년 준에 도착 유행이 일 년 반 전에 시작했어요.

In Korean, the verb <도착하다> is mostly only used for the arrival of physical/visible things, such as the arrival of a plane, parcel, or person. We avoid using it for invisible things like viruses, we would find a way to convey the meaning in a different way, such as saying <the pandemic started> or <the virus is in the country>.

그래서 모두가 실내에서 오랫동안 있었어요.

When using the term <오랫동안>, using the present tense <있어요> gives the sense of the future sense and will be understood as <we are here for a long time (i.e. we will continue to be here for a while)>. We could use the past tense to say <we've been indoors for a long time>/<실내에서 오랫동안 있었어요>.

However, the problem with the correction above is that <오랫동안 있었어요> could also be understood as the past perfect <we had been indoors for a long time (or we were indoors for a long time)>. Therefore, a better alternative would be <그래서 모두가 실내에 있은지 오래에요>.

저는 실내에서 있고 싶지 않지만 백신이 필요해야 돼요.요

<싶다> is <to want to>, and therefore needs to be preceded by a verb, such as <있다>. If you were to say <I don't like being indoors>, you could say <실내에 있는 것이 싫다>.

<-야 되다> expresses the necessity of doing something. For example <먹어야 된다> expresses that you/somebody needs to eat.
<필요하다> also expresses need. <음식이 필요하다> expresses the need for food. <먹는 것이 필요하다> expresses the need to eat, but notice you have to transform the verb to the noun form <먹는 것> before adding <-이 필요하다>.
Because both <-야 되다> and <-이 필요하다> expresses need, you only need one of them, and to use both as in <백신이 필요해야 돼요> ends up meaning <I need to need a vaccine>.

제 언니는 뉴욕에 여름여름에 뉴욕에 가고 싶어해요.

제저는 언니와 함께 갈 거에요.

You can remove <제> as you have specified in the previous sentence that it is your sister you are speaking of.

이건 너무정말 재미있는 여행이에 될 것 같아요.

<I think it will be a very fun trip> as opposed to <This is a very fun trip>.

저는 기대돼요!

If written in full, the sentence would be <저는 여행이 기대되어요!> The reason I removed <저는> (what would seem like the subject), is because when using the verb <기대되다>, the more important noun between the speaker and the trip is in fact the trip! What is important is that you are looking forward to <the trip>, rather than the fact that it is <you> who is looking forward to it. So, unless you mean to say <I am looking forward to the trip, whereas my sister is not looking forward to it>, it is better to omit <저는> if you are also omitting <여행이>.
Alternatively, you can also say <여행이 기대돼요>, although as you have already said that you expect it to be a fun trip in the previous sentence, omitting <여행이> avoids repetition.