Hi Tim, Welcome to the Orchid Board! We're glad you joined us. Columbia is a great town. You picked a really great orchid!
Your plant is a hybrid among genera in the Oncidicum group. A lot of people keep these in sphagnum moss all the time. Others use different media. Some people repot all new orchids right away. I personally would leave an Oncidium that came in bloom in the sphagnum until it is done blooming.
They grow and bloom quickly, so it is unlikely the sphagnum is so old it would cause a problem, and this kind of orchid likes plenty of water, so sphagnum makes sense. Other kinds of orchids run into trouble with being too wet in sphagnum a lot more often than do Oncidiums. I have some in sphagnum; I have to keep Oncidiums very wet or the new growths develop pleated leaves, which is a sign of inadequate water. I can't let mine get even just moist. With my low-humidity growing conditions and heat, I would have a hard time keeping up with their water needs if I grew them in bark, the way many other people do successfully.
You can usually ease an Oncidium in sphagnum out of the pot and keep the root ball intact. This will allow you to examine the roots and medium. If they are good you don't need to repot for a while. Dead roots are soft, brown and mushy. Good roots are firm and white with green tips if new, or brown and firm if older.
Oncidium hybrids tend to begin new growth from the base of the pseudobulb soon after the flowers fade. New roots and a new shoot usually form at about the same time. This is a good time to repot the plant if you want to. I have attached a photo of an Oncidium hybrid I got in October 2015 in spike. It is Cyrtocidium Midnight 'Ebony & Ivory', and its flowers smell very sweetly! It is just beginning to grow a new shoot, the pointy green thing you see at the base of the pseudobulb. No new roots yet.
Choose a new pot big enough for 1 or 2 more growths as big as the one that just flowered. If the current pot has room for another growth, and the roots and sphagnum look OK, I would consider leaving it in this pot until after the next growth blooms. You don't want a new growth hanging over the edge of the pot with its roots dangling, trying to find someplace moist. I could leave my Cyrtocidium in this pot, because the sphagnum is OK and there is room for another growth.
There is a forum of this board devoted to Oncidiums. You can read a lot more about them
here.
And if you use the Search function in the top maroon menu, you can find more threads dealing with your exact plant.
You can also learn a lot at your local orchid society. I see there is a
Central Missouri Orchid Society.
I couldn't tell where they meet, but I bet it's not far from Columbia. You will find information there tailored to your local climate.
Good luck, and we hope to see more of your plants!