A Bard incurs spell failure chance for wearing heavier-than-Light armor, but that wouldn't affect Wands.
Double Wand Wielding requires
Craft Wand and
Two-Weapon Fighting, so as long as you're willing to invest your Feats that way, sure. You've got the caster level 5 requirement set with 5 levels of Bard.
Are you starting at level 6? This build won't be doable until that level, because of the Caster level 5 requirement of Craft Wand. Since you need 5 levels of Bard before qualifying for the Craft Wand prereq feat, and then you'll pick up Double Wand Wielding after that, this build won't work until after you hit level 6 and qualify for your first E6 extra feat. You can be this awesome, as long as you're willing to wait to be this awesome.
You could walk around with Full Plate wielding a single Wand as early as Fighter1/Bard1.
Caster level 5 + any other class 1 could qualify for the Dual Wand Weider build feats. So Rogue1/Bard5 could work.