The party's chairman, Caleb Heimlich, led a news conference on the north steps of the Legislative building at noon, KIRO-TV reported.
"We're demanding Inslee pay back the taxpayers and delivering nearly 5,000 petition signatures to the Governor's Office of hardworking Washingtonians who think it's time for Inslee to do the right thing," a description on the party's Facebook page reads.
Inslee announced Wednesday night that he was ending his bid for the 2020 Democratic nomination.
"I'm inspired by the people I've met across the country. I'm not going to carry the ball, but we're going to make sure somebody is," Inslee said on MSNBC.
Washington State law orders Inslee's Executive Protection Unit of state troopers to escort and protect him everywhere he goes, KIRO-TV reported.
At the beginning of Inslee's campaign, his protection unit was deemed too small to keep up with the cross-country travel, so it was forced to expand from eight troopers and a sergeant to 14 troopers -- two sergeants and one lieutenant.
One hour of overtime for one of those troopers averages about $72, according to documents obtained by KIRO-TV from the Washington State Patrol.
In the month of March alone, WSP troopers worked 1,066 hours of overtime protecting Inslee, and that overtime bill was nearly $82,000.
From March until June, the overtime tab paid by taxpayers was $267,915.
Early in Inslee's campaign, Republican lawmakers objected to the expense.
"I think at the very least, the citizens of this state should not have to pay for his state patrol executive security detail for his campaign," said Sen. Mark Schoesler (R) 9th District.
Several lawmakers attempted to wedge an amendment in the state budget to force the governor to reimburse the state.
"It should be his responsibility when he's out campaigning, to pay the cost of his campaign," said Rep. Ed Orcutt (R) Kalama. But the amendment was voted down.
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