Multiple People in Agency add different signatures on the same document?

Our agency is trying to implement online forms. We have a new hire orientation PDF form that has 12 sections all of which require seperate signatures from Directors. 

Our hope is that the PDF form can be sent out to the new employee and upon compleation of each portion of training the Director can access the document and sign it, and then the next director will sign in etc... 

Can this be done without downloading and saving multiple copies of the same document? Possibily through O365, One Drive or SharePoint? Will the users have to have a specific Adobe program to do so? 

Any helpful tips are apprecaited! 

Thanks, 
Lindsey