The Blue Badge is recognised everywhere in WDW and all the car parks have designated areas for disabled parking although, as TinkTatoo says, you still have to pay. Be aware though, that the Disabled spaces can still be a hefty walk from the park gates and wheelchairs are not available to hire until you are inside the park.
If the nature of the disability is not wheelchair confining then it may be advisable to take one anyway. We got caught out on this with DM's friend who visited with us from Texas, we took her to Epcot and parked in the disabled parking with DM's badges. DM was fine in her electric wheelchair but her poor friend, who suffers from chronic asthma, was really wheezing by the time we walked to the gate and managed to hire her a chair. After a long and exhausting day at Epcot, she really found it a struggle walking back to the car at the end of the day.