No, without a warrant, the FBI is not entitled to search any private web history or documents, and should be held to the same procedure that all law enforcement agencies are. That being said, it is important that these warrants are available in the case of a threat to national security, such as known terrorists operating domestically.
The FBI is wrong to say that they are entitled to access internet browsing history without a warrant. Last time I looked we have a constitution, with a fourth amendment right to be free from unreasonable searches and seizures. Without this protection the government's ability to search you and your belongings would become unfettered.
When it comes to the process of implicating suspects, police don't search homes or crime scenes without getting the legal permission - a warrant - to do so. Therefore, I believe that the FBI should apply the same protocol to internet snooping. FBI officials should request warrants before accessing the browsing history of suspects.
I consider someone's internet browsing history to be private information, much like a journal. Nobody should be able to access this information without the person's consent, or without a search warrant. If someone feels like their privacy has been invaded, they may never feel safe again in their entire life. It is not right for them to access this content without a warrant.