It is one of the few opportunities we get to see some of the best athletes in the world.
Yes, "we get to see ..." is an adjectival clause that modifies "opportunity", but it's missing the word where. The sentence is fine if written:
It is one of the few opportunities where we get to see some of the best athletes in the world.
When might also be fine, or in which.
See this page for more info (scroll down to "WH-Word Adjective Clauses")
Edit: An adjectival clause could be a separate sentence, but as a subordinate clause it modifies one of the words in the "root" sentence. If we break this compound sentence we would get:
"The Olympics is one of [a] few opportunities."
(What happens at these opportunities?)
"(At these opportunities) We get to see some of the best athletes in the world."
The way to construct these is (at least at first) to take two separate statements and link them with one of the wh- words, plus various others like "that":
He has a very interesting story. (In this story) He went to war and came back a changed man.
He has a very interesting story in which he went to war and came back a changed man.
.
It is a vast treasure. (In the treasure) There are mounds of gold and gems within (the treasure).
It is a vast treasure where there are mounds of gold and gems within.
In both of these examples the adjectival clause modifies the direct object (story and treasure). The following modifies the subject:
He is Tarzan, Lord of the Jungle. He (Tarzan) is able to swing from tree to tree.
He is Tarzan, Lord of the Jungle, who is able to swing from tree to tree.
also:
He, who is able to swing from tree to tree, is Tarzan, Lord of the Jungle.
Technically, I think this is no longer an adjectival clause but rather an adjectival phrase because you can't have a sentence that starts with "who". But as I mention, I'm not an expert on the grammar terms.
If we don't consider 'get to see' as a whole the problem is very likely to solve. We might separate 'get' from ' to see' as: we get opportunities to see (using our eyes) some of the best athletes in the world ...
I think you're making it more complicated than it needs to be. In your original sentence the subject is it, referring to The Olympics. The verb is is, and the direct object is opportunity. So this is the root sentence:
The Olympics is an opportunity.
Everything else is just modifying this root sentence. It's not just an opportunity, it's one of a few opportunities. And then we tack on an adjectival clause to tell the reader about these opportunities. A tricky bit is changing an opportunity to the opportunity. If we say the opportunity we are obligated to add more to the sentence to explain which opportunity we mean.
Northern Alaska is one of a few places. (In these places) We can see the Midnight Sun.
Northern Alaska is one of the few places where we can see the Midnight Sun.