How does NEX 5T shoot while charging?

I guess you want to take time-lapse photos?

You can't do this. You can't take pictures when charging, and you can't charge when taking pictures.

It's ok to use an AC adapter, but it's impossible with a power supply. You can also consider connecting a fake battery with a mobile power supply, but there are no related products to sell at present, so you have to find your own way.