Thanks for reaching out with this question.

Some Michigan beekeepers do raise their own queens for splits, although it can be tricky. These beekeepers often manipulate overwintered colonies to reduce crowding and delay swarming, and they often do splits with queen cells. New queens need good flying weather (not to cool or wet) for mating and mature drones for mating, so it's important to wait to graft until the odds are good that the queens that develop will be able to properly mate. There is always uncertainty in Michigan with spring weather for mating flights.

Buying mated queens can be an easier solution for splitting colonies in the spring. Mated queens are caged and are shipped/sold in a screened/vented box with attendants (worker bees to care for the queens). I usually try to place caged queens in queenless splits within a day or two of receiving them, but they can be kept alive for a week or longer if they have enough attendants and food. You'll want to keep the box at room temperature in the dark and keep them alive by feeding sugar syrup. You can feed them sugar syrup by using a spray bottle to spray syrup through the screen. 

You can't put caged queens in a one of your queenright colonies, because the bees will neglect and try to harm the caged queens if they already have a queen. You can set up a queen bank by making a colony queenless and then introducing several caged queens at once, however, banking queens is normally only necessary if you need to keep the caged queens alive for multiple weeks.

Happy beekeeping!