Vilna used to belong to Russia but is now in the country of Lithuania. That is where all your documents will be. (It's possible there will be exceptions but generally documents go with the current government.) Note that some documents may be in Russian instead of Lithuanian. This means they'll be in a non-Roman script, so determining "spelling" is not straightforward.
Doing a quick Google search, I have found that Orsik is a current surname in Lithuania. That doesn't mean the family didn't change their name at some point. Lots of people do, even if they change it from one old country name to another. Also, if the original name was the same but in Russian script, the spelling might have come out different. It's possible that the original Russian ought to have been transliterated to something other than "Orsik" but the family used "Orsik" for some reason.
Immigration officials never changed names but spelling sometimes was a creative process (for everyone in those time periods) and transliterating from non-Roman scripts was always difficult.
Why do you think this family changed their name? Could it not have been Orsik before they immigrated?
If you don't already have it, check out this document. It shows Anton Orsik arriving in New York in 1938. Yes, you said he immigrated in 1913, but this is an Anton who is a US citizen born in 1898 returning from a trip to Europe with his 7 year old son. If the NY address and son's name match, note that the document gives a passport number you can look up. This should lead to naturalization papers.