Is this statement constructed correctly?
So because of their compatibility they were able to work closely together as a strong team should
Particularly, is it OK for it to end in should? If not, what rule is it breaking?
Yes, it's perfectly grammatical. It's an example of ellipsis, in which the verb work is recoverable from the previous clause.
The only 'rule' it is breaking is that of clarity: the one indispensable necessity in writing. Should do or should work are possibilities there, but they are not the same; should alone could be either. The construction has been around, and controversial, at least since the 1930s, when Winston cigarettes used the advertising line 'Tastes good like a cigarette should', and James Thurber suggested that they could follow it up with 'We still make cigarettes like we used to could'.
Language changes, of course, and many people no longer see anything wrong here; how you use or abuse the English language is, as always, up to you. Thurber again: "The written word will soon disappear and we'll no longer be able to read good prose like we used to could. This prospect does not gentle my thoughts or tranquil me toward the future."