“Celestial CFT” is one approach to flat space holography which attempts to recast quantum gravity in (d+2)-dimensional asymptotically flat spacetimes in terms of a d-dimensional Euclidean CFT residing at the conformal boundary. Using this formalism, I will show that continuous spaces of vacua in the bulk map directly onto the conformal manifold of the boundary CCFT. This correspondence provides a new perspective on the role of the BMS group in flat space holography, and offers a new interpretation of the antisymmetric double-soft gluon theorem in terms of the curvature of an infinite-dimensional vacuum manifold.