As the others mentioned, the road should not be an issue. You will be fine.
I'm jealous of your travels. We just drove that route last month. I wish we were headed back that way! You all are going to have a wonderful time. We saw a grizzly bear about 100 yards off the road just inside the east entrance of Yellowstone.
If you are hungry in Cody we ate some unusually tasty BBQ at a food truck. It was sort of on the slower side of town so we found a set of parking spaces to park easily. Check out the reviews: