SOS chaining is really impractical early game. Without the PP, False Swipe, Adrenaline Orbs, Skill Swap, or Leppa Berries needed to extend a chain, it tends to be extremely inefficient.
As you can see in this video, this streamer hunts for his shiny Grubbin via Run Away encounters, which is probably your best bet at efficiently getting a shiny. It does toss up exactly what Pokemon you're going to find, but if this is how you're playing, it's what you have to do.
Or you can just do what Amethyst said and wait till you get False Swipe in Iki Town and Adrenaline Orbs in Route 2, although still at that point SOS chaining is extremely inefficient due to PP.
Hope this helps!