Custom validation to allow only typed input / disallow digital signatures

As the author of that PDF you should consider applying an author signature (aka certification signature) allowing form fill-ins. This makes filling-in of existing fields allowed but changing the form structure (e.g. by adding additional signature field) disallowed.

That should currently prevent that people using Acrobat Reader and similar software can add their own signature fields.

Technically it still is possible to add new fields to the PDF (and consequentially there is software to do so) but doing so will break the author signature and cause messages pointing out disallowed changes to the document in Acrobat. 

Of course people with some knowledge or tools can still strip your form from author signature and any test JavaScript and add their locking digital signatures nonetheless.

The digital signatures are not a part of the text field, so nothing in it can prevent them from being added.

Assuming they are comments you could add a script to the file that deletes them before it is saved. I would notify the users of such an action, though, or they'll be quite annoyed with you for deleting their input without notifying them...

Is there a digital signature field on the form?  If there is, then remove it. 

How are they inserting digital signatures?