The DUP could remove O’Neill from the first minister’s position if the deputy first minister quits.
That role is held by Emma Little-Pengelly, but that seems unlikely.
Over the weekend the DUP leader Gavin Robinson and Little-Pengelly said they would not collapse the power-sharing institutions.
But Lockhart wants that option to be considered.
“There is lots of options on the table with regards to Michelle O’Neill and her continued position, and we have it within our gift to ensure that she is actually removed from that position,” she said.
“So it is time for the anger to be seen in the politics and for profound consequences to follow.
“We’ve been told that all options are still on the table. So that is a very clear option.
“What I am saying is, within the community, within the unionist community, within the Orange community, and within grassroots loyalism there’s a real anger towards a first minister, who claims she is the first minister for all, for thumbing her nose, laughing on the Garvaghy Road last night at this community, who have stood lawfully.”