I have been a postdoc and including occasional lecturer for the last several years. A former student just invited me to his graduation. Would someone at my career stage at an American research university typically be expected to wear academic regalia at a graduation ceremony? I have not seen any specific guidance for this.
It sounds as if you are going to be seated in the audience as guest (e.g., sitting with the family or as if you were friend or family), so just wear 'regular' clothes. It still is a very nice thing to do and to show support.
If you are seated with the faculty (in those universities which have the faculty march in with the students), then those faculty who are part of the ceremony would wear regalia. If you are sitting with the faculty, you probably wouldn't be asking your question, as it is usually arranged in advance to do so (both to make sure there are enough faculty showing up, and to help with renting the regalia for those faculty who do not have their own. And - depending on the size of the university, there will need to be a lot of instructions of where to park, where to mass up and what time to show up in order to march in.).
If the student is asking you to 'hood' him/her (for PhD in many US universities, the advisor can come up to put the PhD hood on the candidate) then talk to the department to find out how to get included in the 'faculty' who are formally seated in the ceremony.
As already noted, if in the audience and not part of the faculty then no one would expect you to wear any academic robes and it would be most unusual to do so.
That being said, when my son graduated from my Alma-mater, where my wife and I graduated and where we met, we felt such a strong affinity to the place and the ceremony that we also wore our academic gowns in solidarity and pride for our son. Although we were the only people in the many hundreds of friends and family in academic gowns, no one batted an eyelid or looked askance. It is likely, for many of them that this was there first time in the audience of such a ceremony and they have no knowledge of the protocol and would just accept anything as normal.
I also noticed the other phenomena, that if suitably robed you can go anywhere...
I face this situation many times as a visiting faculty to several institutions. While most of them accept us as part of faculty and offer cloaks, some accept us as guests only and do not offer cloaks. The only issue I notice in not wearing your academic regalia is when your students wishes to take group photographs with you or introduce you to their parents. You would look alien and un-recognized without regalia for a Academic photograph. This would cause most students to ignore you when taking photographs as well and it can become an insult to your academic profile.