In the arts, a patron is generally called a Mäzen or a Schirmherr(patronage would be Mäzenatentum, but I'd suggest simply using the plural, Mäzene). Or use "Schirmherrschaft", but I think Mäzen fits your context much better.
Gönnerschaft or Begünstigung would actually be incorrect in this context. In fact, Begünstigung smacks of fraudulent practices.

For subscription publishing, I would suggest Subskriptionsarbeit or Subskriptionsauftrag.
Veröffentlichung auf Anfrage, as was suggested, is not very idiomatic in German and does not really hit the nail on the head, especially since "auf Anfrage" is really on request or on demand.

The German word Subskription is often used in the fine arts, for instance in publishing when patrons subscribe to a publication (often books or such) before the works are actually published. My suggestion of Subskriptionsarbeit or Subskriptionsauftrag is based on the fact that you are referring to portraits rather than printed material, and both Arbeit and Auftrag are more general terms.
Subskriptionsarbeit shifts the emphasis to the artist, whereas Subskriptionsauftrag emphasizes the patron (the person placing the subscription order).

Hope this helps.