For some brands like Apple or Nike sellers have a very slim chance of getting approved, as Amazon is extremely strict with those brands. Other than that, new sellers get ungated in almost any brand so long as Amazon is accepting applications and the application meets Amazon criteria. If Amazon is not accepting applications, they will let you know.
Is it harder for me to get ungated as a new seller?
What can I not get approved to sell?
Are there any restictions on brands?