Will Mirror Gallery make it impossible for your Mox Amber to produce mana?
No, legendary permanents are still legendary with Mirror Gallery on the field and will count for Mox Amber.
Mirror Gallery only makes the &quot;legend rule&quot; not apply. It does not change any object in the game. In particular, it does not remove the &quot;legendary&quot; object supertype from anything.
The only way for Mox Amber to not produce mana is if you don't control any colored, legendary creatures or planeswalkers. Note that the game makes a distinction between types of mana and colors of mana. &quot;Colorless&quot; is a type of mana, but not a color, so Mox Amber will not produce any mana if you control only, for example, Karn, Silver Golem or Karn Liberated.

106.1a There are five colors of mana: white, blue, black, red, and green.
106.1b There are six types of mana: white, blue, black, red, green, and colorless.
105.2c A colorless object has no color.