Hello aspirant,
You can definitely appear for the 12th standard exam privately in Gujarat. Generally, you should face no issues in getting admission to a college after passing your 12th exams privately. Your admission will be based on the marks you obtain and the eligibility criteria set by the college.
Different colleges may have specific eligibility criteria, such as minimum marks in particular subjects. Ensure you meet these requirements. Admission to popular courses might be competitive, regardless of whether you've studied privately or through a regular school.

I hope this information helps.