Vocabulary Question

"correct" is an odd word to use, LJ, because it's too loaded a term.

The word order is fine, it's grammatical, but it doesn't sound natural.

I have found a new website fortuitously and have known many new friends from it.

have known - this sounds like you are expressing a finished notion. For this situation we often use "get to know/make/come to know ..."

-> and have got to know many new friends.

-> and have made many new friends.

-> and have come to know many new friends.

I have found a new website fortuitously -

I have found -> The present perfect is used to introduce a past action that has current relevance/hot topic/important to now, so we use it to set the stage and then proceed to discuss it.

I have found a new website. It was quite by chance/accidental/...

Did you use a Chinese-English dictionary to choose 'fortuitously'?

This is what you said:

I have found a new website fortuitously and have known many new friends from it.

This is how I'd say that:

Luckily, I've found a new website and made many friends there.

I stumbled onto a new website and made a lot of friends.

Joe(I'm sure there is more to this story.)Nation

Quite by chance, I came across an unfamiliar website, and now have become friends with many of the members.

Stranger things have happened.

Not sure if it's actually correct or not, but I would change the word order slightly if you want to keep the same general wording.

"Fortuitously, I have found a new website and have acquired many new friends from it.

Thanks all,
I use 'fortuitously' is because I want to learn at least a new word daily.
Is sentence natural?
Fortuitously,I have come across a new website and I have made many new friends.

Pretty much. 'Fortuitously' is not an everyday work, but it sounds fine in this sentence.