Ok, started pretty auspiciously, chose mudkip (named Kipz) and did all the intro stuff. Upon arriving at route 102, I entered the first patch of grass hoping for a ralts. It, however, turned out to be a seedot. Ok, I guess it could be worse, I thought to myself. I was right. With just under half it's life left, I chucked a pokeball. That didn't work, so I thought one more tackle would do the trick. Of course with my luck, that tackle was a critical hit that KO'd Seedot. Needless to say this challenge had not started off on the right foot.
It has gotten much better since then, though. I skipped all the trainers that I could, saving them for later, and worked my way to route 104, where I caught Ashe the Zigzagoon, and Petalburg Woods, where I caught Myco the Shroomish. As of now I'm currently training them up on the trainers I avoided. Petalburg Woods scares the crap out of me right now, and I think I'll avoid the trainers for now, for fear of being poisoned that far from a Center.
My restrictions are no items at all. That means no potions, no status healers, no berries, no hold items. I was going to let myself buy pokeballs, but then I remembered that zigzagoon can get ultra balls via pickup, so that will be my only way of acquiring balls. Yup, I said it.
My team:
Kipz 
lvl 10
Gentle
-Tackle
-Growl
-Mud Slap
-Water Gun
Ashe 
lvl 8
Timid
-Tackle
-Growl
-Tail Whip
Myco 
lvl 8
Sassy
-Absorb
-Tackle
-Stun Spore
I also have a couple questions. Is there some sort of species clause? Like it's possible that you may get a wurmple or zigzagoon in each of your first 3 opportunities, would you only be able to catch one, or multiple? Also, since route 104 is split by Petalburg Woods, are they considered two catch opportunities or just one?