If we were to fail our SCAMPI A Appraisal, when can we try to be appraised again? Who decides, the Lead Appraiser?
Great question! Even though there is technically no "failing" an appraisal, I know what you mean. If there are weaknesses discovered, you can indeed not achieve CMMI MLx.
If there are gaps in the Specific or Generic Goals, it is up to your organization to fix them if you wish to be successful with an appraisal. If it's something simple like, GP2.1 Establish and Maintain a Policy for performing the (x) process, then it might be possible to re-schedule an appraisal quickly. If it's something more complex like GP2.8 Monitor and Control the Process, then the cycle time it would take you to perform this process and demonstrate it through objective evidence could be months, if not years.
The Lead Appraiser is a relevant stakeholder in the decision to re-appraise, but he is not, as our President claims to be, the "Decider." It's really up to your sponsor and your organization.
If you truly believe you are ready to be re-appraised, based on the evidence, then you should go ahead and re-schedule it. If your LA does not agree, that is one good data point in the decision making process, but it is far from the only one.
If your LA is the only one saying you can't do it, and you strongly believe you have the evidence to prove adherance to the model, then you might consider a new LA. That said, they are experts so take this type of decision with care.
There is no "rule" about a waiting period after an unsuccessful appraisal - it's up to you to decide. Of course, failing a second time can be very painful.