Are there Tourism buses from Zhengzhou direct to Yuntaishan?

Yes, you can take a direct bus to Yuntaishan from ZZ Bus Passenger Transport Center (the opposite of the main railway station).
The buses depart at 7:30 and 8:30, and return from the mountain area at 15:30 and 16:30. It costs CNY 38 per person. The trip takes about 2 hours.

In addition, the official website doesn't have English version. So if you have Chinese friends, you can ask help for them. If not, don't worry. Just buy the ticket when you arriving at the bus station. :)

Thank you for your advice. Can we take the 07:30 bus and leave at 16:30?
I presume the fare quoted is a return fare.

Well, if so, the sightseeing time is too short. You don't have enough time to explore the main attractions inside the mountain area. Thus you are advised to spend two days to tour around the mountain.After that, go back in the afternoon of next day. Have fun!