In this paper we construct frames of Gabor type for the space\r\n$L^2_{rad}(R^d)$ of radial $L^2$-functions, and more generally, for subspaces of modulation\r\nspaces consisting of radial distributions.\r\nHereby, each frame element itself is a radial function. This construction is\r\nbased on a generalization of the so called Feichtinger-Gröchenig theory - sometimes\r\nalso called coorbit space theory - \r\nwhich was developed in an earlier article. We show that this new type of \r\nGabor frames behaves better in linear and non-linear approximation in a certain sense\r\nthan usual Gabor frames when approximating a radial function.\r\nMoreover, we derive new embedding theorems for coorbit spaces restricted to invariant\r\nvectors (functions) and apply them to modulation spaces\r\nof radial distributions. As a special case this result implies that\r\nthe Feichtinger algebra $(S_0)_{rad}(R^d) = M^1_{rad}(R^d)$ restricted to radial\r\nfunctions is embedded into the Sobolev space $H^{(d-1)/2}_{rad}(R^d)$. Moreover,\r\nfor $dgeq 2$ the embedding $(S_0)_{rad}(R^d) hookrightarrow L^2_{rad}(R^d)$ is compact.