Are there any safe places to leave your car close to Disney World to avoid the expensive parking fees they charge? ?

No.  There used to be, but not nowadays.

When I go next I'll be flying in, taking the Magical Express between the airport and the onsite resort, and using the buses, the Skyliner and Ubers to get around.  It's easier that way.

If there were, do you really think they'd charge that god blasted much for parking?  Exactly.  

Don't drive there, and you won't have to park there.  Problem solved.

Not unless the hotel/motel you stay at has a shuttle service. If Disney could get away with it, they'd charge you for the air you breathe.

If you can't afford the parking fee, you can't afford to visit WDW

There is no way to avoid the parking fees. You have to get around WDW via Disney transportation system buses, ferries, monorails. Everything is miles from everything else and it's not at all pedestrian or bicycle friendly. 

This is maybe unwelcome, but if you can't stay on WDW property, stay in one of the WDW hotels or campgrounds, then maybe defer that Disney vacation until you can afford to stay "on campus." Then you park for free.

If use your car to commute from an off campus hotel, you'll spend at least a couple hours a day on that short drive to and from. Hours you could be using to "do Disney." And if you stay stay off campus and use a shuttle service, you'll spend even longer on your commute as the shuttle bus calls at hotel after hotel after hotel before finally taking you to WDW. And the bus isn't always there to get you back to you hotel at the end of long long day. 

If you stay at one of the Disney Value Resorts, you will be getting to the parks earlier each day, be able to stay later each evening. Your accommodations will cost more, but you'll get more value out of those costly theme park tickets by having extra hours each day to spend in the theme park. And you won't pay for parking. 

It only costs a little extra to upgrade from a Disney Value Resort to one of the moderately priced Disney resorts. I recommend that highly. It's a big upgrade in room and service for just a little extra money. 

I hope the graphic helps explain why you can't "park and walk" to do Disney. The big circle is Walt Disney World. The red box shows how far is 1 mile on the map. The arrows point to the 4 theme parks. 

Nope.  You'll pay for parking (or get towed) wherever you park near Disney.  You'll also then have to pay for transportation to Disney, which would certainly offset any potential savings by finding a cheaper parking spot.

Some nearby hotels have shuttles going to the park for patrons. Otherwise, no.

Not really. I mean, how would you get there from where you parked? An uber or taxi is gonna run just as much as the taxi