The accusation relates to his efforts to support former President Donald Trump in his attempt to overturn the election result by baselessly claiming that he lost because of widespread voter fraud.
The ad focuses on the Capitol riots on 6 January which left five people dead. It opens up on Mr Trump saying: "Josh Hawley will be one of the greatest champions," at a rally.
Fox News host Brett Baier then asks Mr Hawley in the ad: "Are you trying to say that come January 20th, that President Trump will be President?"
Stumbling over his words, Mr Hawley responds: "Well, that depends on what happens on Wednesday," referring to 6 January, the day when the congressional certification of President Joe Biden's electoral college win was interrupted by the violent assault on the Capitol by a mob of Trump supporters.
Before they attacked the Capitol, Mr Hawley could be seen raising his fist to the crowd in apparent support. Larry Sabato, the director of the Center for Politics at the University of Virginia, tweeted on 9 January that “many senators now despise" Mr Hawley.
In the ad, a news anchor says over violent imagery from the riot: "The Kansas City Star writes plainly that he has quote 'blood on his hands'." Next, Rudy Giuliani yells "Let's have trial by combat!" from the stage at the ellipse next to the National Mall where Mr Trump spoke before the riot.
Another anchor then quotes the former Missouri Senator Jack Danforth who said Mr Hawley’s actions represented the "opposite of conservative, it's radical".
Mr Danforth said in a statement in the days before the riot that "lending credence to Trump's false claim that the election was stolen is a highly destructive attack on our constitutional government. It is the opposite of conservative; it is radical. As one friend asked me, 'What are my grandchildren to think of America if they are told that elections are fraudulent?'"
With dramatic music in the background, the words across the screen at the end of the ad spell out the Lincoln Project’s verdict: "JOSH HAWLEY ... YOU ARE COMPLICIT"
The Lincoln Project was started by a number of communication-savvy anti-Trump Republicans, including Steve Schmidt, who has worked on the campaigns of former President George W Bush, former California Governor Arnold Schwarzenegger, and former Arizona Senator John McCain.
Others include Rick Wilson, who has created ads for several gubernatorial and senatorial campaigns, and George Conway, the conservative lawyer, Washington Post columnist and husband to the former counsellor to President Trump, Kellyanne Conway.
Several ads from the Lincoln Project got under Mr Trump's skin. In December, an ad arguing that former Vice President Mike Pence was backing away from Mr Trump to save his own skin reportedly rattled the then president and made him sceptical of his right-hand man.