I did an arrangement/mash-up of "Without You/I'll Cover You" for a friend's wedding. It was for three female voices, piano, and string trio, so it probably doesn't help you, lol, but maybe it will give you an idea. I'm trying to remember, though, I think that might have been performed at the reception. For the walk down the aisle I arranged "Sun and Moon" for the string trio; that would sound really pretty with solo piano.
There is always the "Processional/Maria (The Wedding)" from The Sound of Music! You're name doesn't happen to be Maria, does it?!
But, seriously: if there is a song that you/your partner love from a show, see if there is a piano version of it on iTunes, or Spotify and listen to it. I'm on Spotify, and there are some lovely piano versions of songs.
"Ok ok ok ok ok ok ok. Have you guys heard about fidget spinners!?" ~Patti LuPone
I happen to be a wedding photographer and I have heard many things used. There was a beautiful ceremony that they walked down to Falling Slowly from Once. It was just a piano to start and then a violin. It was stunning and beautiful.