I've played Vex in tournament with both my Meta Knight and my Pokemon Trainer. I have a very good grasp of how HE plays specifically (his habits, his tricks), but I'm not sure if all the other Bowsers do most of this stuff. When I fought him with MK, I made the stupid mistake of not camping. It wasn't until after the set that I knew camping was the best way to fight Bowser, but the more I mentally analyze the matchup, the more I think it's ridiculously hard for him to beat.
MK can't do his normal poke game on Bowser. Dtilts (more than one, or even only one if he powershields it), ftilts, and tornadoes on Bowser's shield = lol up B. The move does something like 18-20 damage. Not worth the trade. MK's safe options on Bowser's shield are dair, first hit of ftilt (unless they have amazing reaction time), and fsmash. As soon as you hit him, RUN AWAY. Continuing to pressure a blocking Bowser is dumb, since he'll either grab you for oodles of damage or up B you away. Neither of the two are very favorable outcomes.
Since he's going to be hiding in his shield a lot, grabs are extra helpful. Throw in tricky stuff like first hit of ftilt -> walk away to see how he reacts (spot dodge/up B/stays in shield), then just dash grab right back into him. Dthrow to tornado is a guaranteed 20+ damage fresh. Given Bowser's lack of general options after a dthrow, you can regrab him if you can generalize where he's going to land. Once you pop Bowser into the air, his falling options suck, so uair him into nair or up B (or nothing, if he's at high percents), then return to the ground. Don't be stupid and get hit by down B...it's very telegraphed. If you can't move out of the way fast enough, escape it with tornado. He'll either land on the stage with heavy lag or grab the edge. Situation is set back to normal.
When Vex is on the edge, I know he likes to do edge jump fair, edgehop air dodge, and stand up shield/up B/side B. Sometimes he'll regrab with down B. Bowser's ledge attack below 100% is pretty good too. Watch out for all of these moves, as they come quickly and by surprise if you don't know stuff about Bowser. The best idea is probably to stand just outside of the range of his getup attack and let Bowser commit to something, then punish it. If he does edge jump fair, uair or up B the lag on it. If he does get up attack, run and grab it. Air dodge, grab/tornado/ftilt/charged dsmash. It's not hard.
Bowser's vertical recovery sucks, but up B has silly hitboxes on it. Just be careful and hit him with tipper dairs. Two-three should end his stock. If you catch him without a second jump, hog the edge. The reverse up B edgeguard works exceptionally on Bowser due to the lack of hitboxes underneath his up B. If he recovers high, catch him with uair chains/nair/tornado...it's a free hit.
Killing Bowser is annoying as **** if you don't gimp him. Fsmash kills at like 135, dsmash at 160. You can try forcing him to land near you with a charged dsmash for earlier kills. It's important to save this move for obvious reasons. Just remember that every dsmash you used before kill percents could have been an ftilt. He'll be taking your stocks at much lower percents with his grab releases, utilt (don't glide into him predictably), or fsmash (if a retreating fair misses him entirely, he can hit you with this)...but his kill moves are very situational. The most practical way for him to kill you is with a side B (probably will be stale), up B out of shield (probably will be stale), or grab release fair/dtilt. Just don't be stupid. Oh yeah, and don't get Bowsercided, seriously...just fair/uair/tornado him if he's jumping around with side Bs constantly.
Never approach him with tornado...angled up ftilt, utilt, and up B all just **** it. Tornado should be used when he's just slightly above the ground to eat his air dodges, when his shield is low (aim it appropriately to shieldpoke him), or if you KNOW he will spotdodge (not recommended until you know your opponent's habits/patterns).
Synopsis:
Don't approach him, especially in his shield. He can outrange you with ftilt, but it's not disjointed, so you'll win the head to head against it.
Use grabs (dthrow to tornado is amazing) a lot for positioning and damage.
Tornado is the worst approach available, use it properly.
Save your dsmash, it's unsafe on block anyway. Use ftilt as a substitute for damage.
Bowser is screwed without a second jump offstage. If he's below the edge, he's not gonna waste it to attack you or air dodge. Dair him to oblivion. Reverse up B to the edge or stage if he's going to up B. He gets lots of horizontal distance, barely any vertical.
