Yeah, for a single-time trip, I'd enlist your relatives to keep an eye out at goodwill to pick up most of the big stuff, especially snowpants and maybe even boots. Sounds like if you buy gear it wouldn't be used much in Kansas, and kids grow out of stuff so quickly that they would likely never use the coat/snowpants you buy them again.
Wool socks and wool hats are great. They keep you warm even if they get wet/damp (whether from snow or sweat). Light wool under layers are great too, but you aren't likely to find those at goodwill, those are hot commodities.
Get good, waterproof boots. This is even more important than having a good jacket, imo. A jacket wont get soaked under normal circumstances, but boots will. Boots with a wool liner are good at insulating your feet as well. You can add thicker wool socks to "fill out" a boot that is a bit too big for kid feet. I would wear up to 3 layers of socks in my second-hand boots when I was a kid, which was fine because I was mostly crawling around on my hands and feet with my snow projects (so you don't need perfectly fitting boots, bigger is perfectly fine)
People already covered the snowpants thing. Adults probably wont need heavy duty snowpants and could get by with water resistant/water proof wind pants + layers in a pinch. Kids will want to crawl around in the snow though, so you'll want real snowpants for them. When I was a kid I had the overalls type, and wore a jacket over top. Make sure to put the snowpant cuff over top of the boot and "seal in" the boot opening, otherwise you will get snow over top of your boot and it will melt inside your boot.
A good insulated jacket is all you need, and the kids jacket should again be water resistant/water proof. You can add more layers underneath as needed.
Then for the warm accessories - a hat that is warm and covers the ears completely is required (ears are very easily frostbitten), as are mittens. Waterproof mittens for sure, for snowmen and snowball fights and such. you'll want these insulated as well, because hands are the first extremity that your body pulls blood from in the cold and your hands and feet are going to be the first to get numb and miserable. Having them *wet* and cold and miserable is even worse. Mittens are going to be warmer because your fingers can "share" the heat you generate. Gloves with separate fingers get much, much colder, even with high quality gloves. Its just the nature of how your body tries to conserve heat, by sacrificing the extremities.
For a scarf, I would say it is pretty necessary if you are spending most of your time outside. You need something to protect your neck/chin/cheeks/nose from the frigid wind and cold. My mom would make a simple tube of doubled-up polarfleece fabric to wear, along with a bit of a "bib" in front to help keep my chest a bit warmer. These were nice because it was cheap, and they didn't get in the way or fall off, and were warm. You can duck your chin and lower face into it to hide from the wind as you need. Maybe that's something easy you and your wife could DIY.
edit: a trick my mom used to protect our faces for outside was slather on a huge thick goop of vasaline on our cheeks. Windburn is a real concern in the winter, so having this barrier helped a lot. The trick is it needs to be very thick -- vasaline can freeze, and if it freezes to your face the skin can be injured. With a thick layer, only the top "crust" will freeze and leave the rest of the goop to protect the skin underneath.