Native speakers of other languages can absolutely take the CELTA course. In fact, multi-lingual people often make some of the best teachers as they understand the frustrations of their students, and can tap into their experience of what helped them learn the language.
All CELTA applicants have to go through the same, rigorous application process however, so everyone is expected to have a strong understanding of English grammar, pronunciation, spelling, meaning etc. English L2 speakers will obviously not be successfully accepted on the CELTA course if they cannot potentially teach English clearly.
Most English L2 speakers retain their native language accents, and that will not a be a problem provided the accent does not obscure from the communication of the lesson.