Can my iPod Touch 7 generation take Live Photos?

No,

You can take Live Photos on iPhone 6s or later, iPad (5th generation) or later, iPad Air (3rd generation), iPad mini (5th generation), and iPad Pro models from 2016 or later.

Yes, but no.

How to take a Live Photo

1. Open the Camera app.
2. Make sure that your Camera is set to photo mode and that Live Photos is turned on. When it's on, you see the Live Photo button  at the top of your Camera.
3. 
4. 
   
5. Hold your device* still.
6. Tap the shutter button .

Live Photos is on by default. You can turn Live Photos off temporarily or permanently.