When Broker's Hideout enters the battlefield, it sacrifices itself for its ability.
While that ability is on the stack, can I tap it for mana using the ability it gained from Chromatic Lantern?
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Yes, you can use any ability on that fetch land.
Chromatic Lantern grants the mana ability as a static ability, meaning the Hideout will enter the battlefield with the mana ability already on it. The Hideout's sacrifice ability is a regular triggered ability that uses the stack and can be responded to at instant speed, and you can activate mana abilities at instant speed.
Generally, you can interact with the Hideout with any instant-speed spell or ability. That is the very reason why the Hideout has the "When you do" clause on it - the designers anticipated people interacting with the Hideout before it is actually sacrificed to its own ability.