Important: Remember that you will need the Celestial Mirror in order to open the chest with the weapon.
You will find the Spirit Lance on the Thunder Plains, which will require you to complete a little mini-quest with the glowing stones. Throughout the area, there will be stones that will be glowing periodically. You will need to walk up to the stone while they’re glowing and pray at them, which is done with the Square (Playstation)/X (Xbox)/Y (Switch) button. There are two stones in the southern portion of the Thunder Plains, and six in the northern half. All of them are in plain sight, except for the one hidden in a small alcove on the right side of the map in the southern half.
In order to get the chest with the weapon to spawn, you will need to pray at three of these stones. Note that you might be trying to get to a glowing stone and have it stop glowing, so keep trying (especially in the northern half) until you manage to get it. Praying at three glowing stones will cause a Cactuar ghost to spawn in the southern half and following it will lead you to a broken lightning rod tower on the right side of the map. Pray at this broken tower to make the chest with the Spirit Lance spawn and if you have the Celestial Mirror, then you can open it.