First, a simple remark: *If a polytope with congruent facets is inscribed in a sphere, then it is circumscribed about a sphere as well.*

Next, there is a series of examples described and pictured in my old question
https://mathoverflow.net/questions/146763/can-the-sphere-be-partitioned-into-small-congruent-cells .  Each of these examples is what you want in $R^3$. If you begin with any one such example and place it on a great 2-sphere of the 3-sphere in $R^4$ (say, the "equator"), then suspend it from the poles, you will get an example answering your question. The construction generalizes inductively to all higher dimensions.