Can you start a sentence with on?
Since you have provided no information as to the desired contents of the sentence, I assume your question pertains to the issue of beginning a sentence with 'on' in general, while maintaining the grammatical correctness. If that is the case, the answer is YES, one can make a correct sentence beginning with 'on', and here's a suitable example:
'On no account should you let them treat you in the way they did.'
'On no account...' meaning 'By no means...'.
'On what condition would it be possible to rent a car here?'
I am sure there can be many more examples.
All the best,