FWIW, Apple does not manufacture batteries; so, even though they might have sold one, it would be from a different manufacturer even the original battery that was installed when you got the device.
If you do have the original disks, you may be able to boot up with that.
And yes, if it will not even power up, you do need to have it looked at somewhere. Remember that this machine is no longer supported, so unless you are told that the repair would be reasonable, you will need to make a decision if a 2006 is worth spending any money on.