That's right; they are both correct.

Shouldn't it be The more thoughtful your ambitions [ARE] ?

You've almost answered your own question here: the word are is implied, and elided. The listener or reader can infer it based on context, as you and I have.

cf.: The bigger the lie [is], the more people believe it

Or even: The more [people are present], the merrier [those people are]

That's right; they are both correct.

Shouldn't it be The more thoughtful your ambitions [ARE] ?

You've almost answered your own question here: the word "are" is implied, and elided. The listener or reader can infer it based on context, as you and I have.

cf.: The bigger the lie [is], the more people believe it

Or even: The more [people are present], the merrier [those people are]