I'm not 100% sure, but I think that if your car has the electronic door lock option (the switch on the driver's and passenger side), then it should also have the remote enabled. If it is a Sport model, then I would assume that it is more or less a stock feature...does it also have a red alarm light beside the headlight dimmer switch?
That isn't entirely true. Power locks were available as a separate option or as part of a package. If the car in question has the package, then it will have power locks and keyless entry. Otherwise it will only have power locks.
Also, that red alarm light has nothing to do with the keyless entry unless they have the RS security system installed on their car. That is an anti-theft monitoring light for the passive anti-theft system (aka chipped keys), and all it does is tell you that the system is operating correctly or incorrectly.
If they had the RS security system, the light would indeed confirm that keyless entry was present on the vehicle. However, given that the system is a dealer add on and is quite expensive, I doubt that many Corollas actually have it.
To the original poster: No Corolla ever had factory remote start as an option. If you want to see if your car has keyless entry from the factory you can try keylessride. Here's a link: 2004 Corolla Keyless Entry Verification Procedure If you follow that procedure and it works, then you can order a keyless remote from them. They have good customer service, and low prices, plus they give you instructions on how to program the remote. Since there is only one remote for the 9th generation Corolla, you'll be sure to get the one you need.