If you injure another person you're liable for corresponding damages, that's not a function of citizenship.

Nations are willing to assume liability for vaccine side effects suffered by their citizens (which is something that requires resources beyond what an individual could put up - consider the possibility of e.g. becoming paralysed and requiring life-long round-the-clock care). The problem that stateless people have (in this context) is that they don't have a nation-like organisation that will do that for them.