It's up to the judge, but almost certainly not. Judges usually only want lawyers asking the questions. Your lawyer should be prepared, or he shouldn't have taken the case. You may be able to help him by coming up with a typewritten list of questions and expected answers for your witnesses as soon as you can before the trial.
You can also take a notepad to trial and, if you think your lawyer missed very important questions during the examination, carefully and not panicky, clearly print (don't write cursive, which is hard to read) the question or questions on the notepad and hand it to your lawyer. Don't get upset if your lawyer doesn't ask the questions you want him to. Any just-out-of-law-school lawyer can ask questions. One secret to being a good trial lawyer is knowing what questions NOT to ask.