The actual answer is NO, unless you record the video with a device that signs the file with a key that can't be extracted from its hardware. In other words, your videos should be signed by the camera's hardware, which will guarantee the recordings are truly the original ones. Otherwise the files could always have been manipulated by someone or something external, even in real time.
All the other answers are very interesting, but they all have one major flaw: they suppose the video file you provide is genuine, but it's actually your file so you are still in control of whatever you are recording and providing. Between the camera and the upload, anything could happen. A small lag could be added to the video, or it could be slowed down, or looped, etc., even by someone else in real time while you are taking the exam. Actually, if you really want an easier way to cheat, you can just focus on the audio part: it's easier to manipulate the audio in real time (muting the room's channel, mixing white noise or ambient sounds, etc.) to allow someone else to talk to you and help you. When you sign the video with the audio track it's too late, because the result has already been manipulated (gone through a mixer, etc.).
That's why the file must be signed by the device itself: if exam-video.mp4 was signed by Nikon on 2021-01-10 0009:0001:0015, then you can be sure that's the original file recorded by the camera. Note that if the camera didn't detect the difference between the internal mic (original sound) and an external mic (potentially manipulated channel) then you would still have the "audio vulnerability" even in signed files. I actually don't know how such cameras work, all I know is that they do exist, and in the past some were even hacked (someone managed to extract the crypto keys from the camera's hardware).
As a final note, remember that "manipulating videos" and "cheating on online exams" are separate issues. Students can cheat even if they record and provide genuine videos of themselves taking the exam.