The DUP leader said that the “long shadow” of the Northern Ireland Protocol is casting its mark over politics in the region.
Speaking at a press conference with his new MLA team at Stormont, Sir Jeffrey said: “We want to see this place up and running as soon as possible.
“We want stable devolved government. We are committed to our participation in those institutions.”
However, he said that his party’s position had not changed.
“We need decisive action by the Government to address the difficulties created by the protocol.
“Whether that is driving up the cost of living, whether that is the harm that it is doing to businesses and our economy, or indeed in undermining political stability in Northern Ireland.
“The protocol needs to be dealt with.”
He added: “We sought a mandate from people to adopt the stance that we have taken and we will continue, as we recognise others also have a democratic mandate [and] we want to work with them to deliver stable government for Northern Ireland.
“But the long shadow of the protocol is casting its mark over this place.”
The latest comments from the DUP leader suggest that the prospects of any quick return of the devolved powersharing Executive at Stormont are diminishing.
MLAs returned to Parliament Buildings on Monday and party leaders were also holding separate meetings with Secretary of State Brandon Lewis.
It came as Prime Minister Boris Johnson said he does not plan to be personally involved in the Northern Ireland talks and Downing Street played down reports of a Cabinet rift over the protocol.
Thursday’s historic election saw Sinn Fein become Stormont’s largest party.
Speaking to the media on Monday, Ms O’Neill said: “The people have spoken and they have spoken very clearly.
“The message is one of hope, it is also one of optimism for the future, for the political leaders to work together and to make politics work.
“That is my commitment as a political leader and as an incoming first minister.
“The electorate also demands that the parties get back down to business, to elect a speaker, to sit in the Assembly, to have it function, to appoint a first minister and a deputy first minister, to form a new executive.”
“As democrats, the DUP, but also the British Government, must accept and respect the democratic outcome of this election.
“Brinkmanship will not be tolerated where the north of Ireland becomes collateral damage in a game of chicken with the European Commission.
“Responsibility for finding solutions to the protocol lie with Boris Johnson and the EU.
“But make no mistake, we and our business community here will not be held to ransom.”
Speaking following his meeting with Mr Lewis in Belfast earlier, Sir Jeffrey had said that decisive action was needed by the UK Government on the protocol.
Later at Stormont, he said that the sooner the UK Government takes action over the Northern Ireland Protocol, the sooner his party would nominate ministers to join an executive.
“The man with the stopwatch is Brandon Lewis. The people who can deliver the change are Boris Johnson and the government.
“Frankly the sooner they do it, the sooner we can get things properly functioning again here at Stormont.
“That is what I want, that is what I am committed to, but I don’t control the timing of that, the ball is now firmly at the feet of the Government.
“They have talked for a long time.
“They have yet to take the steps that are necessary to address that problem.
“I want the Government to act, I expect the Government to act.
“They know what they need to do, I made that clear to the Secretary of State this morning.”
Sir Jeffrey Donaldson also said that other parties need to respect his mandate.
He said: “I have a mandate, the DUP has a mandate. Our mandate is clear, we want to see political stability, we are democrats, we want the political institutions to work.
“But we re-entered the institutions on the basis of New Decade, New Approach.
“The Government gave a firm commitment to protect Northern Ireland’s place within the UK internal market.
“Two and a half years later they have done nothing to honour that commitment.
“I am holding them to that commitment.
“I am asking other democratic parties to respect my mandate as much as they want me to respect theirs.
“What I will not do, I will not go back on my word. I am not going to say one thing to the electorate and the day after the election do a volte face.”
Asked if he was willing to go for another election in six months time, Sir Jeffrey said: “I am never afraid to face the people.”
The centre-ground Alliance Party made significant gains in the election, increasing its seat count by nine to 17, becoming the third largest party at Stormont behind Sinn Fein and the DUP.
Leading her enlarged team into Stormont on Monday morning, Naomi Long told the media they were turning up for work as they had promised voters during their election campaign.
“I want us to sit down, get the negotiations under way on the programme for government and the budget, and I want to see us getting government up and running as quickly as possible,” she said.
“We’re going to see a functioning Assembly hopefully pretty soon, that will be the easier part of all of this, but to get a functioning executive we need the DUP to step up to the plate.
“With power comes responsibility, and people now need to take the responsibility seriously.
Ulster Unionist leader Doug Beattie said on Monday that if the protocol was the obstacle to a functioning executive at Stormont, then it needed to be dealt with.
“If the obstacle to doing that is the protocol then we need to deal with the protocol.
“We all know what the landing zone is, no checks on goods from Great Britain to Northern Ireland if they are staying in Northern Ireland.”
SDLP leader Colum Eastwood urged the DUP to nominate ministers for a new executive.
He said: “We know who got the mandates, we will support them when we have to, we will also hold them to account, but they should get on with it today and form a government.”
The Stormont Executive has been unable to fully function since February when then first minister Paul Givan resigned as part of the DUP’s efforts to force action against the Northern Ireland Protocol.
Sinn Fein’s election victory means that their Stormont leader, Ms O’Neill, should be in line to become the first nationalist or republican first minister.
However the DUP, as the second largest party, must nominate a deputy first minister to serve alongside her in the joint office.
Meanwhile, the Prime Minister’s official spokesman said that people in Northern Ireland “deserve a stable and accountable devolved government”.
On whether there is any plan for the Prime Minister to get personally involved in the talks or involved in the talks with the EU, he said: “No plans for that.
“This has been led by Brandon Lewis.”