Do all 2003 Elements have remote locks, when I bought mine it didn't come with one, but the locks only lock with the door shut, which made me think it probably did have one at one time?
Have to have a dealer run your VIN to tell you if yours was originally equipped that way.
Bought mine used, and it didn't come with the key fob. did a little research on the net, found where it suggested you go to look to see if it has a wireless lock receiver, then ordered the fobs. 50 bucks, and all is well.
EX's should have them. LX's have a kit available for a dealer installed accessory. Have your dealer check on ISIS to see if yours applies.
I am the original owner of my 2003 Element. The keyless entry was an option. The reason why it only locks from the outside is to prevent locking your keys in the vehicle.