This wheelchair was and is property of S.T.A.R. Labs. Eobard pretended to have been injured after the particle accelerator exploded and started using it after the event, and ditched it after his identity was found out. He used it for two simple reasons, to act crippled in order to deflect or deter any attention or thought of him being the Reverse-Flash until Barry was fast enough to get him home, but also to charge his super speed, using a device that was stored on its underside. He stopped using it after the team discovered his identity.