You can still apply for a Long Form Birth Certificate online. Even though you may not have all the information to fill in - you can still submit the form. I went through and did a trial and it would work.
It does tell you that it may delay the processing a bit longer but she has had a short form certificate previously issued so that may help.https://www.orgforms.gov.on.ca/eForms/start.do
If you need any help filling out the form, I found that they can be quite helpful in assisting you if you just give them a call :)