Given what we’ve cited below, it certainly appears that Senator Chuck Schumer is in direct violation of the statutes and needs to be dealt with accordingly. The FBI is responsible for this type of investigation. However, given their dismal and horrid track record over the past 4-5 years, I’m not confident in their ability or willingness to move forward on any action regarding Schumer’s outburst.

Threatening government officials of the United States is a felony under federal law. Threatening the president of the United States is a felony under 18 U.S.C. § 871, punishable by up to 5 years of imprisonment, that is investigated by the United States Secret Service.[1] Threatening other officials is a Class D or C felony, usually carrying maximum penalties of 5 or 10 years under 18 U.S.C. § 875, 18 U.S.C. § 876, and other statutes, that are investigated by the Federal Bureau of Investigation. When national boundaries are transcended by such a threat, it is considered a terrorist threat.[2]

When a threat is made against a judge, it can be considered an obstruction of justice.[3] Threatening federal officials' family members is also a federal crime; in enacting the law, the Committee on the Judiciary stated that "Clearly it is a proper Federal function to respond to terrorists and other criminals who seek to influence the making of Federal policies and interfere with the administration of justice by attacking close relatives of those entrusted with these tasks."[4]

9-65.700 - Congressional, Cabinet, and Supreme Court Assassination, Kidnapping, and Assault (18 U.S.C. § 351)
Section 351(g) of Title 18, United States Code, assigns investigative jurisdiction over these offenses to the Federal Bureau of Investigation (FBI). The Counterterrorism Section of the National Security Division has supervisory authority over 18 U.S.C. § 351, and should be notified telephonically immediately upon the initiation of an investigation.
[updated January 2020]
(a)(1)  Whoever--
(A)  assaults, kidnaps, or murders, or attempts or conspires to kidnap or murder, or threatens to assault, kidnap or murder a member of the immediate family of a United States official, a United States judge, a Federal law enforcement officer, or an official whose killing would be a crime under section 1114 of this title;  or
(B)  threatens to assault, kidnap, or murder, a United States official, a United States judge, a Federal law enforcement officer, or an official whose killing would be a crime under such section,
with intent to impede, intimidate, or interfere with such official, judge, or law enforcement officer while engaged in the performance of official duties, or with intent to retaliate against such official, judge, or law enforcement officer on account of the performance of official duties, shall be punished as provided in subsection (b).
(2)  Whoever assaults, kidnaps, or murders, or attempts or conspires to kidnap or murder, or threatens to assault, kidnap, or murder, any person who formerly served as a person designated in paragraph (1), or a member of the immediate family of any person who formerly served as a person designated in paragraph (1), with intent to retaliate against such person on account of the performance of official duties during the term of service of such person, shall be punished as provided in subsection (b). (see - 18 U.S.C. § 115 - U.S. Code - Unannotated Title 18. Crimes and Criminal Procedure § 115. Influencing, impeding, or retaliating against a Federal official by threatening or injuring a family member)

https://www.cnsnews.com/blog/craig-bannister/2020-video-schumer-threatening-kavanaugh-goes-viral-after-armed-man-arrested