If you have the Multi-attack feature, or if you have an ability that lets you use the launcher as a bonus action, you can add the extra damage to a single attack on your turn.
Lightning Launcher. A gemlike node appears on one of your armored
fists or on the chest (your choice). It counts as a simple ranged
weapon, with a normal range of 90 feet and a long range of 300 feet,
and it deals 1d6 lightning damage on a hit. Once on each of your turns
when you hit a creature with it, you can deal an extra 1d6 lightning
damage to that target.
A strange note about this extra damage, is that due to the launcher being able to be on your "fists", this means that if you multi-class into monk, this extra damage could possibly be applied to one of your unarmed strikes since that fist has the gem on it.
This is due to the none specific terms used for its activation.
Once on each of your turns when you hit a creature with it
This doesn't specify making any specific attack, only that it hits a creature. This means that this gem, if applied to your fist, will apply this extra damage to any attack, ranged or not, so long as,
you hit a creature with it
If anyone wants to disagree with this assessment, I am just a flawed human, so I can make mistakes sometimes.