Interesting question Ryan. Over the years I have bought a lot of doors and drawer fronts, but never a drawer box as a component. I suspect if you have 6 piece drawers, you might be able to get the boxes directly from the cabinet manufacturer. I have never seen just the box components, (sides, backs, bottoms etc) listed anywhere for sale. You could check with local millwork shops, I'm sure they would be happy to build boxes for you.
The boxes can be made in many ways. Cheaper ones are square cut or simple dido with glue and staples made from particle board etc., while high end drawers are made from hardwood boards, dovetail joints and solid wood bottoms. The price can vary wildly.
If you wanted to build your own, it certainly can be done with a few tools. At minimum, you would need a 10 inch table saw with a decent fine tooth blade (60 to 80 tooth), and perhaps a dido blade. You can make decent simple dido'ed corners and cut a slot for the bottom to slide into. A power stapler and good wood glue would be handy also.
As far as table saws are concerned, I own four of them, one of which is my light weight portable 10 inch with folding stand. I paid less than $200 for it and it has held up extremely well on the job and has no problem ripping up to 2" framing lumber. Mine happens to be an Hitachi, but there are other inexpensive ones to chose from.
Building simple drawer boxes would be a great project for you to learn some basic woodworking skills, get to buy a new cool tool, and have the satisfaction of have done them by yourself. With a little practice on some scrap stock, you will confident enough to build those boxes with decent materials and save yourself a ton of money. Good Luck.