We report vitrimer-like elastomers that exhibit significantly fast stress relaxation using carboxy exchange via the conjugate substitution reaction of α-(acyloxymethyl) acrylate skeletons. This network design is inspired by a small-molecule model that shows the carboxy exchange reaction even at ambient temperature in the presence of 1,4-diazabicyclo[2.2.2]octane (DABCO). The acrylate and acrylic acid copolymers are cross-linked using bis[α-(bromomethyl)acrylates] and doped with 10 wt% DABCO, exhibiting processability to obtain a transparent film by hot pressing. The high-speed bond exchange in the network, validated by stress-relaxation tests, allows quick molding with household iron. In addition, the material is applied as an adhesion sheet for plastic and metal substrates. Because dynamic cross-linking with the proposed bond exchange mechanism can be implemented for any polymer bearing carboxyl pendants, our approach can be applied to versatile backbones, which must thus be meaningful in the practical sense.