Zed's damage output relies heavily on mechanical skill. Before level 6 zed wins trades against veigar in lane using his W->Q or W->E->Q combo. His W->Q shoots out 2 projectiles (one from him and one from his shadow) and if you hit both it can deal a lot of damage early. Veigar is also super weak before level 6 so he really can't contest your presence if you use it to bully him.
After level 6, you can use a combo consisting of W(behind/next to him)->R->Q, which shoots out 3 projectiles that if they all hit, combined with his ult proc, should just about destroy veigar. It's good to use this while his E is on cooldown, as he can otherwise E on himself when you ult him and stun you.
Veigar's main threat is his cage(E), but zed has an ability(W) that just leaves it, leaving veigar with no possible way of hitting his W, and he doesn't win lanes with Q only.
There's probably more to this matchup that I don't know, because I don't really play zed, and if I play veigar into zed I practically just afk under tower all laning phase so I don't die xd