We are hoping for the best results
Yes, the sentence above makes sense. The present continuous tense used in the OP's example, adds emphasis and a certain urgency to the statement, for instance, it might be said at the onset of an experimental medical therapy.
But normally, the present simple is used with hope
We hope for the best results
The present tense with hope expresses future meaning, e.g. She hopes to pass the exam, the exam is set in the future, but the people speaking are expressing a wish, or an aspiration, at that precise moment of time.
We hope the results will be the best
This solution is also possible but it's rather less common than the former, and the meaning is identical.
Cambridge Dictionary says
After hope, we often use present verb forms even when there is reference to the future:
If the OP wanted to express a degree of uncertainty about the results, they could use the past continuous tense, e.g.
We were rather hoping for the best results before making our final decision.
Without further context, it's difficult to know whether this last solution is preferable.