Product keys that can be entered in Steam/EA App are different from the product keys you received when you bought the CDs.
In short, there is no legal way to solve this issue... When you bought the game as CDs, you bought a license to use the software, and you bought the CDs which contained the software. It would still be illegal to download the game installer from elsewhere and use your product key, because that would violate copyright laws. EA is also not obliged to gift you the game on Steam/EA App since it's a different medium.
Basically, yeah, you either go out and get yourself an external CD player for around $15, or you buy the game on Steam for $5 (Steam sales are going on now). But given the situation, there is no way you can get your game working legally without spending a bit of cash. If you do decide to get a CD drive, then I highly recommend you back up the game CDs and store them in Google Drive or something.
