Where can I get the Challenger (normal human tank) in Saints Row IV?

To get this vehicle in Saints Row IV, start a Co-op Cat and Mouse and store it once you have control.

^{Source: Saints Row Wiki, "Challenger - Trivia" article}
Also, the Enter the Dominatrix DLC introduces a new tank, the Dildozer. It is a human tank (kinda) similar to the Challenger. It will appear in your garage once you have finished the DLC missions.

Mods

Random Encounters Mod

Like its name suggests, this mod adds random encounters in the game where you will get attacked by different types of enemies (riding different types of vehicles) at random intervals.

It could spawn enemies that come riding in a Challenger tank (along with other vehicles rarely spawned or not spawned at all [some are SR3 vehicles] by the vanilla game). The tanks can be saved to your virtual garage (default key: N) while driving one.

Screenshots of the Challenger tanks that came with a random encounter caused by this mod:
_{(click on thumbnails to enlarge)}

NPC Behaviour and Spawn Mod

One of this mod's changes are notoriety spawns. At level 3 notoriety, soldiers riding Challenger tanks will begin attacking with this mod.

Note: Some of this mod's files are the same as the Random Encounters Mod. If you want to use them together, either merge those files (instructions here) or overwrite Random Encounters Mod's files with this mod's files.

Additional Cheats mod

OP commented that he doesn't prefer modded cheats, but I'll add this for others willing to use modded cheats and for answer completion. This is probably the easiest way to get a Challenger tank in SR4. Description:

This mod adds 160 new cheats to the game, as well as changing all cheats to non-flagging... That means using them will no longer disable your autosave or achievements.

This mods adds a givechallenger cheat which will add a Challenger tank to your garage.

The only way to get a Challenger in the Xbox 360 version of the game is basically as AcornFuzzyFox stated. However, one needs to make sure that the tank actually drives between the buildings, near where the blue icon to start the challenge is started, in order for said tank to not disappear once the challenge is exited. An easy method of killing its driver without destroying the tank is to get it to maneuver into place, between the building and the bar, then fire a black hole onto the wall a bit to the front of and as high up the building as you can. This will suck the driver out of the tank and not suck up the tank. You will know you've killed the tank driver when you can walk left and right without the gun barrel following you around. It can be a bit tricky to get the placement properly, but it will work if the tank is close enough. Remember to save the tank by pressing down on the D-pad until the purple car icon is filled.

Bonus tip: Also, yes, you can get a Bear APC using this method on this same challenge, too. The challenger can't be modified at all, but the Bear can be modified using the "Rim Jobs Glitch". 1: After saving the Bear (or any normally non-modified vehicle, like a flatbed), exit the vehicle and proceed to the nearest Rim Jobs. Once there, press the button indicated by the keys icon. This will take you to your Garage menu. 2: Retrieve a modifiable vehicle (such as a Compensator). This will put you in the garage area, driving said vehicle. 3: While still in the garage, press the button indicated by the wrench icon. This will take you to the Rim Jobs menu. 4: Choose Garage. This will take you to your Garage menu. 5: Highlight a "non-modifiable" vehicle (such as the Bear). Then press the button to exit the garage (not the one to select the vehicle). 6: You will be prompted with a question: "Are you sure you want to exit the garage?". Choose Yes. This will activate the glitch and take you to the Rim Jobs Menu, displaying the normally non-modifiable vehicle. The Bear can't change colors, but it can get new wheels and/or performance options, including nitrous and kneecappers. 7: Enjoy having a stronger, faster, and all around better "non-modifiable" vehicle.