This is obviously false in the case that an error is made in the process. E.g. someone vaccinates themselves with a live virus, or possibly accidentally engineers an even more virulent strain.
If you are making your vaccine from artificial peptides that are a hundred times shorter than a live virus, this seems unlikely. There was no way for live virus to be introduced into this process.
The parent said "Homemade vaccines are not contagious (just stating the obvious). ", they didn't say "homemade vaccines made from artificial peptides are not contagious".