Say I have a Mycosynth Wellspring on the field and a Piston Sledge. The equip cost for piston sledge is "sacrifice an artifact". If I control no creatures, can I sacrifice the Mycosynth Wellspring to the Piston Sledge to benefit from its ability, or does the equip ability require a valid target in order to pay its cost?
No you may not, you need a target to activate the equip ability. Equip means from the Comprehensive Rules
Activating an ability is very similar to casting a spell, and most of the same steps need to be followed to activate an ability. If you cannot comply with all the steps, then you have performed an illegal action, and the game backs up the game state to just before that illegal action.
This is the important step that you are incapable of complying with. You cannot announce a valid target for the equip activated ability.
BTW, if you did have a single creature that the Piston Sledge was already attached to, you could sacrifice an artifact to Equip the Piston Sledge to the already equipped creature. If the creature was an Artifact Creature, you could sacrifice the creature to pay the costs of the activation since costs are paid after targets are chosen.