Interesting question Heather!

It is in fact not an adverb but an adjective /attribute used with a verb of state (vivre).

In this particular example, the verb être ( which is a verb of state par excellence!) is implied as -

vivre (étant) heureux = to live your life happy ( rather than happily) it describes the state of being happy.

You could say-

vivre caché = to live a hidden life

Other verbs are 'demeurer',  'rester', 'mourir'...

Il est mort jeune = He died young

Elle est morte pauvre = She died in a state of poverty

Ils sont resté coincés là = They remained stuck there

It is a tricky one to explain, I believe it is the same in English, and I hope this helps!