At the moment, Donald Trump is deliberately refusing to allow anyone he has any control over to testify before Congress. In typical Trumpian fashion, he completely fails to understand that doing so only makes him look more guilty, but he is stupid and entirely ignorant of matters of law and legal procedures.
They have various mechanisms at their disposal to coerce administration officials to testify, none of which they have yet used.
Still under discussion is the question of whether Robert Mueller, the special counsel whose report sparked the current battles between the House of Representatives and Trump, will testify.
But the House should also look further afield. Trump’s first Secretary of State, Rex Tillerson, has stated publicly that Trump asked him to undertake illegal actions while he worked for the administration. He is no longer a federal employee and presumably has no particular loyalty to Trump. The House should invite him to testify.
At the moment, a law suit by the State of Maryland and the District of Columbia, claiming that Trump violates the foreign emoluments clause of the Constitution that prohibits public officials from accepting gifts from foreign leaders is still alive in the federal courts. The House should invite attorneys who are working on that case to testify. No matter what the federal courts decide about allowing the case to continue, the House does not operate under the same legal constraints. The emoluments clause essentially prohibits bribery under another name, and bribery is one of the crimes the Constitution lists as grounds for impeachment, so the House can make a strong case that violating the emoluments clause is grounds for impeachment.
The people who work for the House Judiciary Committee could likely come up with quite a list if they sat down and thought creatively on the topic.
They should find a way to impeach Trump. He deserves it and the country needs it.