That doesn't sound right. Chevron doesn't mean the government can do whatever they want - the court always has the authority to step in and say "Hey, that's unconstitutional," when it is actually unconstitutional.
But by definition, Congress can't delegate authority it doesn't have. In particular, Congress doesn't have the authority to transgress the Constitution.
So there's no question of whether the statute permits the agency to transgress the Constitution. It never does and never can. Even if the law said "Congress authorizes the CBP to violate the Constitutition," the agency would still not be authorized to violate the Constitution.
In the next year I would imagine there will be a movement to overturn what the CPB has done.
Do you have one source for this? A case where someone tried, or a blog post by a legal expert discussing this?
As others keep explaining, the Chevron deference never allowed agencies to defy the law. It merely allowed them to have the final word on how an ambiguous law that applied to them should be interpreted. For example, if Congress passed a law saying "there shouldn't be poison in the water", the EPA would have the final word on whether a particular pesticide at a particular level constitutes "poison". A company couldn't sue and say "we think this is actually harmless". But the EPA couldn't say "this law authorizes us to hang anyone that puts poison in water" and get away with it through Chevron deference.
If something CBP does is challenged on its constitutionality the court doesn't need to go to Chevron, it will say that the enabling legislation is unconstitutional.
Chevron was essentially the decision that an agency established by congress to make rules for a domain had rulemaking powers for that domain so long as it doesn't go against express congressional decisions.