Twitter hasn’t accepted verification requests for any users since April 2019 and that now looks to change. Currently, you cannot apply to be verified and the page simply says ‘Check back soon!’.
Jane Manchun Wong, reverse engineer, has now discovered evidence in the Twitter Web App that the company is looking into bringing this feature back to all users. The screenshot suggests Twitter will add an option in the Account settings to request verification.
It’s not known when Twitter plans to bring the feature out or release it publicly at all, but it’s a sign that they are working on a new way for users to become verified. It’s currently impossible to manually ask for verification but Twitter has verified random users when they become popular.