In the US, if you commit a crime, it nullifies immunity. If Phizer knew the vaccine was dangerous, lied about it, and promoted the vaccine, then they're not immune from a lawsuit. Because criminal behavior and intent is bad faith.

If Pfizer pulled the vaccines as soon as they discovered they were unsafe, they would be immune. The Color of Law Act makes it a crime for any state actor or quasi state actor to knowingly deprive people of constitutional rights. CNN was once found liable as a quasi state actor because the incident involved CNN working with a government agency. If a private actor is assisting government interests, then they're public actors and can be sued for censorship and deprivation of rights.

Redditors constantly circulate the myth that private entities are protected from lawsuits if they censor people or violate people's constitutional rights. That is absolutely the arrogant mentality of corporate lawyers and most government actors. But it's incorrect. People don't bother looking up the actual case law.

So corporate owned America media companies have been found liable for deprivation of rights despite being a private entity. Just Google private actors color of law or quasi state actor color of law and there's tons of legal journal articles about the topic. Erwin Chemerinsky has a free Coursera class (or used to) about constitutional law. Highly recommend taking it. It covers how to make a private individual/entity liable for constitutional violations.

So if Pfizer was funding vaccine campaigns knowing it was dangerous, they're not immune. If they censored vaccine victims, they can be held liable since Pfizer's interests were the same as the governments.

Canada has something akin to common law but I'm not familiar enough with their tort law to comment.