Q.green carpet rupturewort
We are interested in replacing our small lawn with this lawn substitute. Since this plant has a taproot (how long I do not know) and we have gophers who eat roots throughout our hills, is this plant’s taproot at risk?

Green Carpet Rupturework or Herniaria are not considered 'Gopher resistant'.
Addressing the Gopher issue prior to planting would be the best coarse of action.
Here are some articles that have more information.
https://www.gardeningknowhow.com/special/xeriscape/drought-tolerant-lawn-grass.htm
https://www.gardeningknowhow.com/lawn-care/lawn-substitutes/green-carpet-lawn/herniaria-green-carpet.htm
https://www.gardeningknowhow.com/plant-problems/pests/animals/eliminate-gophers.htm