Hello Carl,
Thank you for your question about using cardboard printed with ink for sheet mulching. The following articles discuss this topic:

Can I Use Cardboard With Ink In My Garden?
Black ink is typically derived from soybean oil. Allow the material to become the primary mode of transportation. A colored ink that is used on paper and cardboard can contain toxic heavy metals. Colorants found in cardboard or paper should not be used for mulch.

Gardeners’ opinions differ on the use of cardboard in vegetable gardens. Certified organic growers can use what I call ‘clean’ cardboard – plain, unwaxed boxes that have all of the tape and sticky labels removed. When a site is treated with cardboard as a surface vegetation, the trauma is less than when it is dug up. Because of the rise of e-commerce, we are now receiving an increasing number of cardboard boxes in our homes. If you live in a dry climate, cover cardboard mulch with compost or another material to keep it moist. Termites are occasionally seen eating cardboard mulch that is too dry.
Source: Don’t Use Printer Ink For Gardening: It’s Not Safe For Plants – AccessibleGardens

Is Newspaper Safe for Your Garden? - The Permaculture Research Institute (permaculturenews.org)

If you need additional information, I recommend you call the Deschutes County OSU Extension office in Redmond at<personal data hidden> and ask to speak with our community horticulturist, Amy Jo Detweiler (oregonstate.edu). Amy Jo welcomes your home garden questions and is a great resource. She may have more information on this topic.