The rule that stops you from using Chop Bot without a target is the “Using Abilities” rule (emphasis mine):
Any time an ability is optional, regardless of whether it is a triggered ability or a constant ability, then the player is considered to be “using” that ability and the card it is on if they choose to resolve it. A player can only use an ability if its effect has the potential to change the game state. This potential is assessed without taking into account the consequences of paying any costs or triggering any other abilities.
So you have to have a target for it when you choose to resolve it. That’s the only time that the game checks this. But when exactly is it that you choose to resolve it? For that, let’s look at the “Trigger Met” step under “Trigger Steps,” which is where the detailed timing of abilities is explained (emphasis mine again):
Trigger Met: The trigger condition of a conditional ability has occurred or the trigger cost of a paid ability has been paid, and the ability is ready to trigger. If a paid ability or an optional conditional ability enters the “trigger met” state, the ability is considered to have been “used”. Note that multiple conditional abilities can have their trigger met at the same time. Whenever one or more abilities are in this state, a player must trigger one of them. If the active player controls any of the abilities, that player chooses one of their abilities to trigger; if not, the inactive player chooses one of the abilities they control to trigger. After that ability is triggered and resolved, another ability in the “trigger met” state (if any) should be triggered in the same way.
So the choosing, and thus the check for a target, doesn’t happen until after the card is already in the “trigger met” state. Therefore, it can get into the “trigger met” state without there being a target. But it does have to be installed, as you said, so that it can see the trigger.