Yes, people give birth standing up all the time!
It does make things a little harder for the midwife - I have certainly gotten a crick or two in my neck from trying to get underneath someone giving birth, but that’s kind of how midwives operate. Since we support and respect normal physiologic birth, we don’t tell people to get in bed or on their backs just so that we can see better.
It can be a little wacky in the immediate aftermath - that is, getting everyone comfortable, making sure baby isn’t yanked too hard, making sure the new parent doesn’t slip and fall in amniotic fluid or blood, etc.
People give birth kneeling, on hands-and-knees, in a lunch position, in a squat, and in other upright positions all the time. More graphic and not-graphic photos of birth below the cut.