The future events, the ones 9 months after Alex Parrish started training, revolve around terrorism in New York City. After the first half of season 1, Alex is assigned to the New York field office. The key points of both is that the FBI is in charge of terrorism prevention and after-attack events.
The reason for the FBI's involvement with the Vice Presidential Candidate and the Presidential Routes is that the FBI leads the Joint Terrorism Task Force:
A Joint Terrorism Task Force (JTTF) is a partnership between various American law enforcement agencies that is charged with taking action against terrorism, which includes the investigation of crimes such as wire fraud and identity theft. The agencies that a JTTF comprises generally include the Federal Bureau of Investigation, other federal agencies (notably Department of Homeland Security components such as U.S. Coast Guard Investigative Service, U.S. Immigration and Customs Enforcement, U.S. Customs and Border Protection, the Transportation Security Administration, and the U.S. Secret Service as well as the Department of State's Diplomatic Security Service), state and local law enforcement, and specialized agencies, such as railroad police.
JTTFs engage in surveillance, electronic monitoring, source development and interviews in their pursuits. FBI task forces obtain written memoranda of understanding (MOUs) between participating law enforcement agencies. The FBI provides funds to pay for participating agencies' expenses, such as officer overtime, vehicles, fuel, cell phones, and related office costs.
In the Show
Before the Bank Bombing, the FBI is there because of the Grand Central Bombing, while the Senator/Vice Presidential Candidate is also in NYC. It's a coincidence, so the FBI/JTTF ups security.
Post bombing, FBI Agent Wyland was the FBI liaison to Senator Hass' Secret Service detail in the JTTF, and thus has access to the Secret Service information. She was later removed as liaison due to her takedown of the unidentified EMT at the mall and the incident at the Town Hall in s02e05 "Clue". This is why Alex used Wyland's fingerprint for access, and why FBI Agent Wyland even had access in the first place.