You've not provided much in the way of your available equipment and skills, and the task you've set for yourself has substantial supporting requirements.
A book press is top of the list. You can build your own, with appropriate research, tools and materials, or you can purchase one to fit your needs. A book press is used to compress paper pages, but can also be used to apply pressure to the book board into the print surface after ink is applied.
Do you want to print and bind the pages yourself? That's a separate question from the title question.
You would not be able to print a "board" page with a conventional paper printer, which then requires something like old-style letterpress type. If you do a search on eBay, for example, using "wood type" you'll get a lot of (possibly expensive) results.
On the less expensive side for this form of printing is linoleum block printing. This can be fun, as it can be more free-form if you've any talent as an artist. The block surface is cut away where printing is not done (in reverse) and inked, pressed and peeled.
The link above points to an Instructable with some very well done artwork.
To add a touch of high tech, one can laser engrave special rubber in a similar manner. This results in precise, sharp designs of any intricacy and detail, but adds complexity and expense.
In the process of collecting these images, I was amused to see how many of them are not mirror images. It's important to have them flipped, otherwise the print becomes reversed!
$20K will buy a whole lot of silk screen supplies which can do full color with proper photosensitive screen layout. You can do CMYK separation with almost any graphics editor, even a free one.
If your images aren't too complex, screen printing will work, be less expensive and possible more fun if you're doing it with family members. You print an image on inkjet film, layer it on a photosensitive screen, expose to light, wash away the unexposed areas, then force ink through the mesh onto the board.
The multicolor aspect does make things more interesting, as you'd have additionally registration marks and photographs would be out of the question, but blocky full color cartoon/comic type images would be easily accomplished.
Quite a project you've presented. I wish you all the best.