How can I fix this handheld vacuum cleaner?
Easy fix if it's the batteries

The batteries are 18650s they are extremely common they are used in power tools, laptops, home devices and they are even the driving force behind the Tesla automobiles.

Test

Cut the red and black wires going to the batteries one at a time. Strip off a little bit of insulation and apply them to a 9 volt battery. Red+  black -

Leave the yellow wire for now. It senses the voltage for the charge indicator light. It's possibly connected to a chip wrapped in the battery casing.

This is what the individual batteries look like

CAUTION

These batteries have a lot of kick. They are potentially very dangerous if the contacts are shorted. They will melt wire and are very capable of starting a fire. 

If the unit runs on a 9v then the batteries need replacing. If that's the case you will need to order some Amazon has lots of them. You'll want to find some with the same mAh rating. In your case you will need two 1300mAh batteries. 

something like this, basically apply power directly to the motor.