With Zangief his ultimate goal is obviously to grab you, and secondly is to get close enough to you so that you fear his grab and make decisions in fear of his grab. This leads to the wakeup options you’re having trouble with. That’s Gief’s domain, and you’re likely going to be at a disadvantage if you find yourself in it, as he’s played well enough that you’re he’s forced you into playing his wake-up game.
With Vega (Claw), you can abuse your good range to zone him. Keep him away and it’ll frustrate him. Try to bait his V-Skill with a safe poke and then fish for a hit to chunk down his life. I don’t play Vega truthfully so I can’t help you with exact normals, but I imagine that’s the main idea.
When it comes to wake-up options, it’s really all a more complicated game of Rock Paper Scissors. It cycles between: backdashing out of corner, neutral jumping, jumping away or blocking. Depending on what the Gief does, these are usually your options.