I've been doing versus a lot on Hal's stunt track, and I found that most power-ups can collect coins. But which ones can or cannot?
Red's Boost? No.
Stella's Big Bubble? Obviously. Her bubble can collect coins from about two feet away from her kart's radius.
Bomb's Explosion? Yes.
The Blues' Boosts? No. Boosts don't seem to collect coins.
King Pig's Balloons? Too high to collect any coins.
Terence's Thunderbolt? Yes.
Bubbles' Balloon Form? He can get big enough to collect coins with his power-up, so yes.
Matilda's Egg? No.
Foreman Pig's Dynamite Sticks? No.
Hal's Tornado? Like Stella and Bubbles' power-ups, he can collect two spread-apart lines of coins.
Corporal Pig's Spinning Helmet(s)? No.
Chuck's Chuck Time? No.
Ayrton Senna's power can't do it, either.
If any observations are wrong, please leave a reply below.