Syringing your ears is best done by a doctor to prevent damaging your ears because if not done correctly then syringing can lead to problems with your ear and possible hearing loss.
You can however do syringing of ears at home if you know how to do it properly so you don't damage your ears.
Syringing your ears should be done as a last resort though when you can't remove the earwax buildup by any other safe method.
To syringe your ears use a ear syringe irrigation kit that includes a bulb syringe to clean your ears out.