That's a common sentence structure. There's no reason an adjective can't follow a verb. The sentence could be rephrased:
A Thai youth soccer team and its coach were alive when they were found Monday...
A Thai youth soccer team and its coach were found Monday, alive, in a vast, flooded cave complex.
"Alive" refers to the state of the team and coach, it doesn't modify the verb "found". There is flexibility in where it can be placed in the sentence without changing the meaning, as long as it's clear who "alive" refers to.
"Found" also refers to the team and coach. You could say:
A Thai youth soccer team and its coach were found Monday...
and not talk about the state of their health. The fact that they were alive can be added to the sentence. Putting it after the verb doesn't change what it modifies. "Found alive" in the referenced sentence just means the team and coach were found and they were alive (both words referring to the team and coach).