I am using a TinyZero with the Tiny WiFi Shield to create a web host on my local network. This lets one use any browser to query and control the TinyZero. For example, turn on a LED or a NeoPixel (a WS2812B). This works well when the TinyZero is connected to USB for power, but not when it's solely on fully-charged battery power (150mAh lithium-ion)  The LED will still turn on with battery power, but the NeoPixel will not turn on.

USB Power: LED and NeoPixel
Battery power: LED only

Why is this happening?
And how can I illuminate one or more NeoPixels without requiring a connection to USB power?

This is simply because the NeoPixels require 5V to operate and the Li-Po battery only supplies 3.7V. USB power would supply 5V, but if you want to power this project from a battery, look into a higher voltage battery or battery pack.

Hope this helps.