Yes, but you'll need outside help if you only have a GameCube and a GC memory card.
20XXTE can be downloaded and placed on an SD card, which Wiis are able to read. The files can be transferred to a GC memory card plugged into the same Wii. And from there, it can again be transferred to other GC memory cards plugged into the same Wii or GC. This requires that you know someone else with a Wii and SD card who's willing to help.
The other alternative that's no longer available was to pre-order a 20XXTE memory card - a GC memory card with the data already on it, but those are out of stock. Although you wouldn't have been able to get TE updates without eventually having to use the above method anyhow, so you're not missing much here. TL;DR find a friend who's willing to help, or go to a local tournament/check your local facebook group and ask around to see if anyone has 20XXTE files on them.
Edit: Regardless, I would highly suggest looking into getting the 20XX 3.02 training pack, which is separate and apart from TE. As it is, TE doesn't serve much of a purpose for now; it's additional features were kept to a minimum so that it could be easily accepted as tournament legal, but even in the latest update they're still buggy. Until they're fixed, TE shouldn't be tournament legal anywhere. The catch about the 3.02 training pack being that it's only available via Wii or Dolphin (computer emulation), but the money or time put into obtaining either will be well worth it.