It's not possible to cover all Mexican holiday foods, but a few of them are true classics. The number one dish is tamales, probably because they're very time-consuming to make (like many special holiday dishes). Tamales are steamed masa, or corn dough, formed around a filling (typically meat or cheese) and wrapped in either a corn husk or a banana leaf, depending on the region where they're made.
There are plenty of shortcuts to be found, but they won't be whole foods. Masa harina should just have corn and lime as its ingredients. Most recipes will call for lard, but they may also use butter or oil. One unique way to make a healthier but still whole-foods version is to use cooked, mashed plantains for the dough. For meat fillings, find recipes that call for cooking from whole cuts instead of using something like rotisserie chicken from a deli. You can roast or boil a whole chicken and then make broth, which many recipes will require. The same goes for beef fillings. Dried corn husks are available packaged, but you could also save corn husks and dry them yourself. It's a lot of work no matter what, which is why tamales are usually made by families in huge batches.
Now let's look at a type of bread that doesn't include corn: Rosca de Reyes, or Three Kings Bread, baked specifically for the Christmas holidays with a Jesus figurine inside. Whoever gets the figurine is supposed to be in charge of making the tamales for the next holiday. While the dough itself is a pretty traditional yeast version, one of the special aspects is the candied fruit. Around the holidays, it's usually easy to find candied cherries, lemon, orange and other fruits. But they contain high-fructose corn syrup and preservatives, so you can actually candy your own fruits. They're essentially cooked in a sugar syrup and then dehydrated.