I am a private candidate of 12th.What should I fill in neet application form as there is no private option?

Candidates who have passed 10+2 from Open School or as private candidates shall not be eligible to appear for ‘National Eligibility Cum Entrance Test’.   Furthermore, study of Biology/Biotechnology as an Additional Subject at 10+2 level also shall not be permissible.

The proviso in italics has been subject matter of challenge before the Honble High Court of Delhi, Honble High Court of Allahabad, Lucknow Bench and Honble High Court of Madhya Pradesh at Jabalpur. The provisions of the regulations disqualifying recognised Open School Board candidates and the candidates who have studied Biology/Biotechnology as an additional Subject has been struck down.   The Medical Council of India has preferred Special Leave Petitions before the Honble Supreme Court and Appeals in the Honble High Courts.

Therefore, the candidatures of candidates in the NEET (UG)-2019 who have passed the qualifying examinations i.e. 10+2 from National Institute of Open Schooling or State Open Schools or as private candidates from recognised State Boards; or with Biology/Biotechnology as additional subject shall be allowed but subject to the outcome of Special Leave Petitions/Appeals filed by the Medical Council of India.

This is what is mentioned in the pdf of the rules as per the website.
YOU ARE ELIGIBLE.
You can get further information from the NTA helpline