No groundcover is "dog-friendly" with respect to handling regular or rough-use foot traffic (human or pet). (If you're also asking about a dog occasionally eating the plants, we don't have toxicity information and defer to the expertise of a vet.) Turfgrass is the one of the most resilient plants to cover the ground where it needs to hold up to dog running, urine, and/or digging, and even then, repairs via overseeding are often needed every year.

Some of the species in our Groundcovers web page plant lists might hold up reasonably well to foot traffic, but stems can be crushed (so, for instance, flowering perennials might not bloom well) and the roots may become stressed from the soil compaction, which is not easy or quick to alleviate/remedy. We don't have research to refer you to regarding non-turf foot traffic tolerance, though we know some people are looking into studying that trait in the future. Nothing will be immune to "burn" damage from urine, though that is at least temporary if the plant is able to replace the injured foliage and roots.

Miri