You can find areas to camp along the Delgermurum River, which is situated less than 2km from Mörön, but, being near to the road, it's not the most peaceful place to visit in Mongolia. There's all sorts of accommodation in the town and, unlike most other places outside Ulan Bator, it has a large market and supermarket!
It has to be said that grocery shops in villages on the steppe are pretty much the same. You'll be able to stock up on staples: vodka, beer, hot drinks, biscuits, sweets, noodles, sardines, salad and, occasionally, frozen meat, potatoes, a couple of onions, maybe a carrot or two, if not a stray turnip! So best to buy supplies in Mörön, where the supermarket resembles Alibaba's cave.
You'll also find shops along the main road in Mörön where you can pick up fishing tackle, foldable rods and lures at rock bottom prices. There's an interesting, modern wrestling stadium and, having seen matches in Ulan Bator, I recommend you buy a ticket. You'll also find a provincial museum with exhibits that include shamans' ceremonial robes and ancient ornaments.