Per our blog post:
Eventually, all developers with existing access to our APIs will be required to complete a developer account application in order to maintain their apps. We will provide at least a 90-day notice before enforcing this requirement. We may opt to retire apps.twitter.com before enforcing that requirement; in that case, we will ensure developers have access to manage their apps on developer.twitter.com, even if they have not yet started the application process.
I’m sorry to hear that your application was denied. Please do know that we are going through each and every application manually to see if your use case matches take into account our developer policies. If you were denied, we found that your use case either directly conflicted with our policies or that you did not take the time to fill out the use case with a meaningful case. At this time, there is no way for you to reapply, but we will be updating the process flow to allow such an action in the future.