It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back $EG=BG_{/\ast}$ gets you the Kan fibration you wanted.
A Kan fibration over a 1-category is a biCartesian fibration in groupoids, and being G-principal is a condition on the fibres.