Our song is Do You Remember by Jack Johnson. Here is a link, if you would like to listen:

http://www.last.fm/music/Jack+Johnson/_/Do+You+Remember

What type of dance would you suggest for this song?

I realize that it probably won't be anything particularly romantic or elegant like a waltz, but it has been our song for years so I would really like to get it to work with something.
Btw, my fiance and I are beginners, having taken about 6 months of basic ballroom classes.

Thanks in advance for your help!

I listened to your song and it is great. It would be a perfect Rhumba for your wedding dance.

You could do a Bolero or slower Latin Rumba, but the song rhythms don't go with either dance. You could also do a country Two Step, but the rhythms don't really work with that either. The song is not really a dance tune.

You've chosen a song that does not have a proper timing or bpm for a Waltz or Fox Trot. It certainly does not work for a Latin style either.
A good teacher may be able to create a routine of some sort that could work. My suggestion would be to look at the video of the Fox Trot's Sway Step on this site in the Learn the Dances section and see if incorporating that step or a similar pattern might not be a good start. That step in closed position and then maybe breaking apart with a turn, while keeping in time, could work with some creativity and practice.

Any ideas given to you in this forum are valuable, and I would certainly try some rumba steps to this dance -- if you know the rumba and are very comfortable with it. I couldn't hear a rumba rhythm in this song, so I personally wouldn't try it. Based on what you said about being only six months into ballroom, the slower dances will be very difficult for you to do with the balance and fluidity they call for. I still think it will have to be a dance choreographed for your skill level and the occasion -- so it won't be any particular ballroom dance, per se, but a few steps, turns and features of maybe a couple of different dances.
Good luck. It will be fun, no matter what.