"...nothing appears to be broken."
It didn't just fall off. I think a trip to a Canon service center is in order. They will make sure nothing is broken.
The T5 battery door is not designed to be easily removed like other cameras because Canon did not design a battery grip for it.
There are some 3rd party battery grips available for it though, and the 3rd party instructions say to pull on the door at a 35 degree angle to remove it.
To re-install the door I would just reverse the process and gently push on it at a 35 degree angle.
Thanks everyone for your suggestions. Ultimately I followed Ray-uk's idea of searching YouTube, and sure enough I found plenty of videos demonstrating how to remove the door. None actually showed how to put it back on, but after seeing how easily it came off, I had the confidence to apply a little pressure and it popped back on easily. There are actually two spring loaded clips that you can press down with the door hinge, leaving more space for the hinge to slide in.