For long time, the community has theorized they were siblings1, but RIOT never confirmed it, until Arcane.
When Jinx was first announced ( during September 2013, which was after both Caitlyn2 and Vi were released on January 2011 and December 2012 respectively) this came with a message in the Vi's champion description on the league of legends website, wich was "vandalized" by Jinx. It is called "X WUZ HERE". There, Jinx insults Vi in various way, making fun of her voicelines in the game.
Shortly before Jinx release, another post came on RIOT forums, this time it was Vi addressing Jinx and swearing to stop her - with a punch in the face, as usual. The post is called "Mayhem Everywhere You Look".
When Jinx was finally released, we had a look to her splash art, which resembled the Vi's one, but destroyed. This has been further confirmed by RIOT itself - this message popped up when they introduced the messages in the loading screen.
First, we can observe that Jinx has some dialogue lines made especially against Vi's, mocking her catchphrases, theme song and such3. But then, there is one line which started everything.
"You think I'm crazy? You should see my sister!"
Assuming Jinx is telling the truth, this line states that she has a sister, and since the close relationship between Jinx and Vi, they were speculated to be sisters for a long time.
This speculation reached RIOT, which in return, made another in-game joke, this time with Hermit, the shopkeeper in the Howling Abyss map for the red team.
Here, if you are paying as Vi, the shopkeeper will occasionally says
"You look like your sister! Er, wait, I'm not supposed to talk about that..."
Vi on the other hand have only a minor cosmetic buff if there is an enemy Jinx, called "Catch me if you can!"4.
On last note, it's important to note that question about Vi and Jinx being sisters can be found even before Jinx release!
On October 3 2013, 10 days prior to Jinx release ( but after X WUZ HERE, Mayhem Everywhere You Look ) the so-called Jinx team posted an AMA on reddit. One of the hottest topic was indeed if Jinx and Vi were actually siblings, and RIOT Ransom said on the matter:
Not sure why this is such a hot topic. Vi and Jinx don’t behave like siblings at all! I mean, they’re always bickering and making fun of each other, and that’s not very sisterly, right? Actually, wait a minute… Perhaps this topic DOES merit some further consideration. So here’s what we know: Vi claims in Jinx’s teaser that she has no idea who Jinx is or where she came from. So if these two are related, Vi clearly hasn’t gotten the memo yet. Which is interesting, and there’s also the fact that we don’t know a whole lot about Vi’s past as a street urchin. So who knows? Seems like there are a lot of interesting story paths this rivalry could go down. I can’t wait to see how it will play out!
1: While I can't research for every single post regarding this theory, after a fast search I found out one post dated July 2014 on reddit wich dig in the question.
2: Caitlyn has some connection with Jinx as Vi, but Vi seems to be a lot more closer - one way or the other - to Jinx based on much Jinx "target" Vi instaed of Caitlyn.
3: There is a lot here.
Jinx's quote "She's such a loser, always ready to cry!' T-t-t-ta!" mocks Vi's theme song first phrase. Jinx's quote "Jinx? Stands for Jinx! Durr." mocks Vi's quotes "Vi stand for Violence/Vicious/Vice". Jinx has also special taunt lines against Vi: "You're gonna chase me, fat hands?", "Vi? Stand for stupid.", "I'm over here, fat hands!"
4: The buff itself is just an icon wich keep count of how many times Jinx killed an enemy Vi and/or Caitlyn or how many time a Vi or Caitlyn have killed an enemy Jinx. On second note, is important to remember that Vi voice over never received an update ( writing this on November 2021 ) and since Jinx was not yet conceived, she had not special lines with her.