Just wonder, when you rent the wheelchairs, is the process long in the am? We don't want to spend a long time waiting each am (and wasting our extra magic hours) to get one for my father so I'm contemplating bringing one. However, if you can just get one and go quickly then I thought I might do that instead of lugging one on the plane, etc
There are agencies that you can rent from that will deliever and pick up from your resort. I know people do this with strollers and scooters. Perhaps you may wish to consider a scooter, as pushing a wheelchair can be taxing or so I have been told. Whichever you do find a way to make it stand out for the times in which you will get out of it.
The answer is "it depends". Wheelchair rental is at the same place a stroller rental, si I'd there are a lot of people renting strollers it can take some time. The actual process isn't long.
That being said, I would STRONGLY recommend an ECV, especially if you're staying on property. Last trip I hurt my ankle at Typhoon Lagoon and was in a wheelchair the last two days of the trip. DS12 pushed me around, but AK was brutal with the his and rutted paths. Epcot wasn't much easier, being so spread out. If you're staying on property, as @Jedilogray said there are a number of companies that will deliver to the resort, and they're cheaper.
I rented wheelchairs for my daughter on two trips and don't remember it taking more than 10 minutes (this was 8 and 11 years ago though). I am going next week and am thoroughly disappointed with the misinformation I've gotten from cast members today about wheelchairs. one told me i could take the park wheelchair to the Boardwalk area (definitely not true!); another told me I could have the hotel wheelchair for the parks and elsewhere (didn't believe this to be true); then i called the hotel who told me they no longer provide wheelchairs, but backtracked that there were a few to get to and from rooms (not sure what is true!)! i guess i will have to rent offsite (didn't want to have to transport it to and from the parks, especially when we drive) but because we will need it for the Boardwalk, there seems to be no other option since there are no rentals there. If you only need it in parks, i don't think you will have a problem (though high attendance days, they do run out).
When we were there in July and had to rent a wheelchair we were told they were phasing them out soon. That may be why you are getting little/conflicting information. There were still some around, but few and far between at the resort (YC). The one we borrowed to get around the resort actually was supposed to be from HS and was a bit beat up, but it got the job done.
We've rented wheelchairs twice on our trips and it was always quick...5 minutes or less (this was September, so low crowds). We bought the whole week at one time, so after the first day we just gave them a sticker (I think it was a sticker) and claimed a wheelchair. If you're there pre-RD, you can get them while you wait.
In our situation, the person needing the wheelchair could walk enough to get to transportation and around the resort and it was definitely nice to not have to worry about transporting the wheelchair from resort to park. We didn't park hop on those trips, so I'm not sure if the rental transfers from one park to another.
We did not have any issues with pushing the wheelchair. Again, September crowds were low, so we didn't have to fight that part, but the hills weren't too hateful.