The short answer: It depends on who you ask.
The U.S. Constitution provides sparse detail on the impeachment process, and there has been little opportunity to establish precedent because impeachments are so rare (only two U.S. presidents have been formally impeached). Therefore, due process rights during impeachment are ill defined.
What we do know is that due process is an important legal concept that goes all the way back to early English common law. It is also, of course, found in the Fifth and Fourteenth Amendments of the Constitution, providing that no person shall be deprived of life, liberty, or property, without due process of law.
Impeachment is another concept found in the Constitution, primarily in Article II, section 4. It gives the House of Representatives exclusive power to impeach an official, and it makes the Senate the exclusive court for impeachment trials.
Impeachment is limited in scope and can only remove an official from office (and disqualify that official from holding future office). It cannot assess fines or order jail time for crimes — that is left up to the American judicial system.
This brings us to one side of the argument, which is that presidential impeachment is a political process, not a legal one. For that reason, a president going through impeachment does not have due process rights like they would if they were going through a civil or criminal trial.
Yes, there are many similarities between the impeachment process and the criminal court process: there is evidence, a trial, and a decision rendered. But, the impeachment process is not a criminal process — despite the similarities — and some constitutional scholars argue the protections for an official going through impeachment are extremely limited.
Both presidents who have faced impeachment in recent history — President Nixon and President Clinton — have been afforded their own legal teams and an opportunity to defend themselves in the process, but not until the process moved to the House Judiciary Committee.
For example, legal teams for Nixon and Clinton were able to play a role during the House Judiciary Committee's hearings, including asking questions, suggesting witnesses, and requesting evidence. President Trump's legal team will be able to do the same.
However, there are important hearings and investigating that takes place before the process moves to the House Judiciary Committee, especially in President Trump's impeachment process. Trump's legal team has not been invited to participate thus far, and Republicans have called the process unconstitutional and a violation of the president's due process rights.
Those who believe that due process rights do not apply during impeachment argue that while the Constitution does drive home the concept of due process of law (it's the only command that is stated twice!), the protection applies when a person is in jeopardy of losing "life, liberty, or property," none of which apply in impeachment when the only risk is being removed from office.
Even if due process protections do apply, other constitutional scholars say, they would not apply until the trial, which takes place at the end of the process with the Senate acting as a jury (the Constitution tells us this much, at least).
Up until the process reaches the trial, and the House is conducting investigations and deciding whether to bring formal charges. It is akin to the police investigation or grand jury trial in a criminal case, some say. It's not until trial that defendants are given the opportunity to defend themselves, call witnesses, see evidence, etc.
Unfortunately, like so many matters pertaining to the constitution, there is no simple explanation of a president's due process rights during impeachment. At the end of the day, we know that due process rights are not necessarily guaranteed to a president going through impeachment, but if they were, they would likely apply at the trial step of the process.