The sentence is something that you might see as a formal exchange between institutions, but even then it is slightly unusual. I would reformat something like this "I sincerely wish that our student exchange program continues, and that the friendly relationship between our cities continues to grow and deepen."

Do you mean the use of "be"? It's perfectly correct. You're looking at Present Simple Subjunctive" - an action that is wished-for, demanded, requested, and so on. Here's a link which explains it a little: http://www.tolearnenglish.com/exercises/exercise-english-2/exercise-english-91954.php