I'm gathering from your question that you are wanting a location that you aren't having to book ahead of time and are just hoping to make the vow renewal just happen.
While I can't guarantee that it will be 100% private, I highly recommend a location at the Polynesian Resort. On the beach of the resort is an area called Sunset Point where the beach extends out a bit into the Seven Seas Lagoon. A path takes you to the top of the point; it is a magically romantic area where weddings ceremonies have even been performed.
Similarly, along the path that runs in front of the Boardwalk Resort's path to Epcot, there is a "gazebo" area that overlooks Crescent Lake. Again, super romantic!
If you love the Animal Kingdom, there is a "secret" path that leads guests down along the bottom of the Tree of Life. This area rarely has guests around it and would likely provide you with some privacy. However, it is tricky to describe how to get there, so you'd have to ask a cast member at the park to help you find the specific path.
If you'd like a more guaranteed amount of privacy, you might want to consider book a
fireworks cruise for you and your family. It can be done with either the Magic Kingdom's Wishes or Epcot's Illuminations, but while I don't imagine a problem, I'd encourage you to confirm when you book the cruise that they'd allow you to have the vow renewal.
Wishing you a magical vow renewal!