I've come to the conclusion that "tirar" has its standard photographic meaning here of taking (i.e. exposing) the photograph itself. Tirar, hacer and tomar are all used for this, of course.

It's quite true that "tirar" can mean print:
"14. tr. Impr. imprimir. Tirar un pliego, un grabado"
Normally, however, it means this in the field of printing rather than photography. I don't think it's out of the question for "tirar" to be used for printing copies of a photo, but it's not really usual.

In any case, supposing it does mean print, the fact is that you don't print negatives, you develop ("revelar") them; you take prints from negatives. "Tirar el negativo" seems to me an unlikely way of saying the latter. And if it were used with that meaning, we would be left with a tautology; it would make no sense to speak of "copias de época" (vintage prints) being printed at the time the "negative is printed" (prints are taken from the negative), since making the vintage print is itself printing from the negative.

An Internet search is no help here; there are no relevant instances of "tirar el negativo". But that in itself is revealing, since if "tirar" were a technical term for something you do to or with a negative you would expect to find instances.

Thus I'm driven to the interpretation that involves a standard use of "tirar" and that also makes sense: a vintage print is one made at the time the photograph is taken. Actually, of course, the negative has to be developed first; that part is taken for granted. The text conflates exposing the film (shooting the photo) and developing the negative into "shooting the negative", which in English is a perfectly normal expression:

"So the first print produced from a negative is simply an educated guess as to what the photographer saw when shooting the negative."
http://www.pfsphoto.com/FAQ.html

The relevant definition of "tirar" is therefore this:

"13. tr. Fotogr. Disparar una cámara fotográfica."