No. Half the point of LLM's is having them research topics on the internet as your agent. It's totally harmless in the usual (tool-based search) workflow.
(Would we also need a real ID to operate curl(1)?)
Regulations should attach only at the point humans willfully or recklessly cause their agents to commit crimes.
But keeping with the analogy, a car manufacturer would be liable for a faulty design that caused wrecks, no? It doesn't need to be willful by the user.