Yes, yes, yes! EVs and IVs make a huge difference in competitive play. At level 100, your Pokémon can gain up to 84 extra points in a stat (before applying nature) if you’ve maxed out your EVs and IVs. That’s easily the difference between getting outsped and OHKO’d vs outspeeding, dealing damage, and then surviving the hit that would have OHKO’d you without proper EVs/IVs (or just outspeeding and OHKOing the opponent before they can attack at all). If you want to get into any level of serious competitive play, you need to be using Pokémon with optimal EVs and IVs.
Edit: when playing ingame, yes, your Pokémon need to have the proper EVs and IVs, the game will not auto-set them like it will to your level (since there’s no way for the game to know what EVs you want unless you set them yourself).