Yes.
They do plenty of indictments on their own.
They can't. A grand jury issues an indictment.
The FBI is not in control of the process once a grand jury is empaneled or subpoenaed.
If the grand jury concludes that there is probable cause to believe that a particular individual committed a crime, the grand jury will issue an indictment.
Whenever a grand jury is involved in an investigation, the agents will work closely with an attorney from the U.S. government, either from the local U.S. Attorney’s Office or the U.S. Department of Justice, before making an arrest in order to determine whether a crime was committed and, if so, who is responsible.