Like the title reads, I was just wondering whether or not I need a link monster in my deck to play Yu-Gi-Oh! at tournaments or my local game store.
Using Link Monsters is optional.
It's the same case if you decided to, say, build a deck without any Trap cards.
If your deck or strategy doesn't require them, you can play without having to have one.
However, if you plan to play in tournaments, or play competitively, I'd suggest you include or at least considering using Link Monsters, as your options could improve (and because the meta game is tending towards using Links).
You don't need them. A reason someone might say they 'need' them is because their strategy relies on swarming cards from the Extra Deck.
They're absolutely optional, but I highly recommend looking into how the new Master Rules affected the decks you're currently playing, and see if Link Monsters would help your strategy.
There are still decks in the format that work fairly well without any cards in the Extra Deck, like True Draco, for example.