Arduino standalone with 9V battery and accelerometer
Your problem has one of three probable causes. 

Cheap or dead 9V battery.

Current draw too much for a 9V battery.

A bad linear regulator on Vin.

Try a new 9V, and not a dollar store brand. If that doesn't work, try a 7.5 to 12 volt adaptor on Vin. This will help eliminate the three probable cause.

Usb bypasses both the linear regulator and avoids the low quality current issue of 9V batteries. A cheap solution for Lipo battery packs are using usb battery packs, they are lipos with charging circuitry, cutoff and boost to 5V. No mudding of the pack or Arduino needed