Le stress n’est pas quelque chose de mesurable, car chacun le vit différemment

The meaning of "vit" there is "to experience".

Let me try to explain the grammatical structure of the bolded part by a progression:

Chacun vit (everybody experiences)

Chacun vit le stress (everybody experiences stress)

Chacun vit le stress différemment (everybody experiences stress differently)

Chacun le vit différemment (everybody experiences it differently — the object being understood from context)

From a grammatical point of view because who asked, considering “chacun le vit différemment” :

• chacun is the subject
• le is a direct object (referring to le stress)
• vit is the verb
• différemment is an adverb (precising how people “vivent le stress”)

And in fact, the key element of this sentence is the coordinating conjunction “car” which binds the second part of the sentence to the first as a cause.

About the meaning or translation, @qoba is right.

The structure is

Subject + Complément d’Object Direct + Verbe + Complément circonstantiel de manière.

Note that : “adverb” is not a grammatical function ( but what french grammarians call a ” word nature” : noun, adjective, adverb, pronoun, etc. ). The ” nature” of a word never changes. Its grammatical function varies depending the sentence in which it is used.

Chacun –> pronom indéfini

le –> pronom personnel “il” in the accusative ( direct object) case ; here this pronoun has as referent the noun phrase ” le stress”

vit –> verbe

différemment –> adverbe de manière