If you are going to be doing something obviously a wedding, then you are going to have to go through Disney if you do it in any public area. Now say you want to take a more secluded area in a resort, join hands and say a couple of nice words to each other (keeping out of the way of other guests) and then take a few photos with the kids, you could probably do it. But if you wanted something formal with flowers and a photographer, well, that could be an issue. Disney is quite protective of it's wedding department. You could rent the gazebo out through Disney Catered Events (probably) for a dinner or something, if you wanted to go that route, but I'm not sure what your budget would be.
People always mention the Swan/Dolphin as an alternative, but I've looked and for a party your size, you are not saving any money at all going through them as opposed to an onsite Disney vow renewal.
I'm celebrating my 20th this year as well. We did a Wishes VR for our 10th and I can't swing another one for this year, unfortunately. We're going to Victoria and Albert's, but I don't know if that'll work for you as you have your kids along.
Some other ideas include ordering a cake (there's a cake thread in the restaurant section), booking a fireworks cruise (that's super romantic, we did that after our vow renewal), taking a carriage ride, maybe sending the girls off on their own for the night (since they're teens) and going out to dinner just the two of you? There's romantic restaurants in several price ranges. I found it really romantic to stroll the Boardwalk/Yacht/Beach Club areas at night after dinner. But the beaches in the Grand Floridan/Polynesian area are nice as well.
Congratulations!