Preface: The lack of a in-game VOIP for Splatoon sucks.
Did Call of Duty: Black Ops 2 for Wii U have voice chat? Doesn't Monster Hunter 3 Ultimate have it? I thought one of them has it, otherwise why would Turtle Beach make a headset for the Wii U.
On the assumption that one of the games above has voice chat and that they run on Nintendo Network servers, there isn't any hardware or Nintendo Network limitation on there being voice chat. For Splatoon, it appears that the developer didn't care enough to have the feature in their game.
I... can't say that I entirely disagree with that assumed lack of care. Most multiplayer shooters these days have players live and die on their skill to read a mini-map, especially in games with short time-to-kills that Splatoon appears to be. Combined with the low 4v4 player count, I don't predict you'll run in to many situations where voice chat would have saved your squid from being splattered. For a player providing covering sniper fire in the game's ranked King-of-the-hill mode, the low player count mitigates the lack of voice chat as there are less targets for the sniper to worry about and less chances of a team's front liners being outnumbered.
To be fair, I can imagine various situations where voice chat would've been a boon. Coordinated flanking maneuvers that push cleverly and safely in to the oppositions territory. Crucial firing angle placements or death call outs that could stop a dominating enemy squid's kill streak. While a mini-map shows a lot information with skilled reading and the low player count means a smaller amount of information to parse, voice chat can provide information and tactical orders that mini-map reading and pre-set messages cannot.
Also, you can't talk trash with your friends, which is something I know first-hand can make an online multiplayer game stick around longer for a person.
While I, personally, am not too disappointed by the lack of in-game voice chat in Splatoon and think it's absence isn't as damaging as others think, it's a shame it's not in there.
Here's to hoping it's in the sequel!