Troy,
If you're considering building a Sonotube-based sub, you owe it to yourself to thoroughly investigate the
Sono Tube Projects Link at the top of the DIY section.
Patrick Sun built 3 different "Sunosubs" that you should definitely look at. On his
first, he did use vertical bracing (wood), but not on the
other two. Some have made threaded rod work for screwing on top and bottom endcaps while running the full length of the tube as well (
DustinB did something similar).
Personally, I don't see the need for vertical bracing and I didn't use it on my Sonotube sub. Tight-fitting endcaps, glue, and caulk worked great on mine.
It's your project, though. Do it how you feel comfortable. That's one of the benefits of DIY.
Good Luck!
JKS