Hi Bridget
I'm sure you will be fine. I did this tour last September and the coach left Cody for Yellowstone pretty early in the morning to be there when the park first opened to avoid the queues. It will probably be busier over a bank holiday weekend but all of the shops and restaurants are geared up for large influxes of people.
My only two points of concern would be trying to find a good spot to view Old Faithful when it blows (every 90 mins) because it's always packed and people start grabbing the seats in the viewing area up to an hour before - a bit of a waste of time if you ask me. Personally, although I saw it, I didn't think it was as spectacular as I had imagined and frankly, your time at that stop would be better spent wandering up the hill and viewing some of the other beautiful silica pools and geysers. I liked the Bee Hive Geyser which is not so reliable but is really impressive - I hope you get lucky and see it blow.
My second thought is that they mentioned to us last year that there were plans for a major renovation of the bridge that connects Cody to Yellowstone. The work is well overdue and the delay has been caused by working out the logistics of rerouting the main road to the park while it is closed (it could take a couple of years to do the work once they start and an alternative route could take buses a couple of hours out of their way). I don't know what the current situation is but I'm sure all of the tour bus companies have been kept informed and will have contingencies in place which I'm sure JY will have been advised about. Don't worry about it though, the drivers are all aces at adapting quickly to any situation - ours used weather reports to his advantage on our second day there and got us in to the park half an hour earlier than planned which was brilliant because we just missed a heavy fall of snow which closed the gates behind us. A number of other buses got left behind and had to wait several hours for conditions to improve while we practically had the park to ourselves.
Have a super time and happy to help if you have any other questions,
All the best
Mandy