All the time on The Perils of Penelope Pitstop. Every single episode. More than once. Funny thing is she has only been gagged once in "Carnival Calamity". About 90% of the Hooded Claw's schemes probably would have worked if he hadn't overlooked that.
April O'Neil in the 1987s Teenage Mutant Ninja Turtles, who always ended up being bound and gagged by the bad guys, especially during the earlier seasons, to the point that the turtles can even recognize her mumbles.
In the Turtles Forever special, the 1980s Turtles tell their 2000s counterparts that they must save April at least once a day.
Not as often as you might think for a character nicknamed "Danger-Prone Daphne," but regularly enough to deserve an honorable mention.
Happens in most episodes of Batman: The Animated Series to one character or another. Most notably, Catwoman finds herself on a Conveyor Belt-O-Doom being carried toward a cat food grinder in the episode "Almost Got 'Im", but this is far from the only example.
In SpongeBob SquarePants we get a very brief glimpse of Mr. Krabs and Patrick tied-up and helpless in the storeroom, courtesy of our favorite sponge as he prepares a fancy dinner for Squilliam and his buddies; it was the only way to stop their disastrous attempts to help.
There's another incident where Mr. Krabs is tied to his chair but not gagged as SpongeBob and Squidward torment him to see if he's really a robot. It Makes Sense in Context.
In Hey Arnold!, Harold, Stinky and Sid come to Arnold for help after thinking they blew up a police station When it was actually Ernie who demolished it with a wrecking ball. When Arnold tells them they have to go to the police, they bound and gag him for fear that he would rat them out.
The U.S. Dennis the Menace cartoon once had a scene where a seal is running amok, and at one point grabs toilet paper in its mouth and runs around the kids, wrapping them up in toilet paper. They even get gagged by apples that get knocked off a table and fly into their open mouths.
This happens to Brattus in the third act of the Mr. Bogus episode "Bogus To The Rescue", after Bogus busts him for making a false alarm on the phone about an emergency.
This happens a lot in The Simpsons. One notable example is in Cape Feare, when Sideshow Bob ties up the entire family (even the pets!) as he goes off to kill Bart. Of course, Homer is asleep throughout the ordeal.
Lisa: Oh, no. Dad's been drugged!
Marge: No he hasn't.
This sometimes happens to Princess Toadstool and less frequently to Mario and Luigi in the Super Mario Bros. cartoons. Even the Koopa Kids get tied up once.
Caleb from "WITCH" gets caught, bound, and gagged in season 1. Lucky for him, he'd done the same thing to the same guards earlier in that episode with his belt, where he kept a dagger that he used to escape.