Does that mean the father has died or that the child was born out of wedlock and might not know who the father is?
The application is in Monroe County Ohio in 1902.
My guess is mother refused to name the father or didn't know his name. If it was criminal activity, (my post wouldn't budge with the correct word used), she might not know who it was.
I agree with Gail that "no legal father" is most likely a reference to illegitimacy. If the child were half-orphaned, they would've said so: "no living father" or some such.
You have received two very similar responses that are very good suppositions regarding what is listed on an Ohio marriage application. You might also want to review specifics about Ohio records by reviewing information in the following Wiki article.
It is often amazing what details we can find when we search deeply. Often there is information connected to a record that has not been indexed or well explained on the actual document, but where the details are part of the record but were not required to be indexed.
I totally agree with the above comments that the most likely reason to see "no legal father" would be related to a mother who for whatever reason failed to name the father at the time the child was born. Best wishes in your continued research. Since this question is really about searching in our Historical Records, I will be moving the post to the Search category in the Community Help Center so that others can find the discussion if they wish to comment.
Thanks for the responses. As I have continued the research on the individual, I have found a registration of birth document where the child does list the father. However I cannot find any marraige record for the parents nor does he show up on the census record one year after the birth of the child. This further supports the idea that the child was born out of wedlock.