Is there any Boosting algorithm with closed-form solutions?
Basically: no.  Boosting is a "meta-method".  If you have a weak learner (an algorithm for learning), it is a way of taking advantage of that weak learner algorithm iteratively to do better at learning some concept than the weak learner would do on its own.

Many/most weak learners don't have a closed-form solution for the model they build; instead, their learning algorithm may fundamentally rely upon iterative methods, optimization algorithms, or other techniques.  As a result, boosting with such a weak learner won't have a closed-form solution, either.