I can't speak to OpenAI's specific setup, but a lot of startups will use a third party service like Carta to manage their cap table. So there's a website, you have an account, you can log in and it tells you that you have a grant of X shares that vests over Y months. You have to sign a form to accept the grant. There might be some option to do an 83b election if you have stock options rather than RSUs. But that's about it.