Its a very doable drive..
Julian -- as Midwest Michael noted, the most scenic and direct route will be close in January. That's a 2 lane road which goes over the Sierras and gets heavy snow in the winter -- so it closes usually in about November and doesn't reopen until sometime after April.
The alternative routes you can take are to go west from Vegas across the high desert around the southern Sierras up to the western entrance of Yosemite through Bakersfield. All very good roads, and I've never heard of this route being closed in the winter. (I've seen snow on the high desert -- a couple of inches overnight -- once.) Its going to be a day's drive though to do this trip, but Yosemite is open year round and is very beautiful in the snow.
Another alternative route it to go north along the eastern side of the Sierras up to Lake Tahoe/ Reno area, and then down over Donner pass into the Sacramento area. From there you would turn south to come down to Yosemite from the north west. This is a major interstate highway and the major pass over the Sierras and is a very high priority to keep open in the worst weather. I've heard it being closed, but never more than like overnight. It is a longer route though -- figure 2 days at least, particularly if you stop in Death Valley on this route. But this route could take you through Death Valley, Lake Tahoe, Sacramento, and possibly the California Gold Rush country on the way to Yosemite.
Once you make it to the western side of the Sierras you are in the large California Central valley -- which doesn't get snow. So no problems again until you get near Yosemite and go back up into the mountains. The roads into Yosemite are mountain roads so in a major storm they may be closed for a day -- but the plows will be out pretty much immediately clearing the roads into and out of the park.