Are there any molecules with delta bonds in their ground states?
Chromium(II) acetate has a Cr-Cr quadruple bond - and hence a $\delta$ bond - in its ground state. This sort of "paddlewheel" configuration, with four carboxylic acids coordinating a pair of metal ions, is reasonably common (see also copper(II) acetate, for instance), although the extent to which the metals can be considered covalently bonded varies.