The tense of the construction habe sein dürfen is the perfect tense in conjunctive I mood; it uses an Ersatzinfinitiv rather than the more systematic sein gedurft habe — which, as a side note, sounds horrible to my ears. In indicative mood, it would probably be transformed into a preterite form in a newspaper:
Nichts macht mich stolzer, als dass ich eure Vorkämpferin sein durfte.
The speech in video form and a transcript can be found on hillaryspeeches.com. I suspect the passage in question is the following:
And to all the women, and especially the young women, who put their faith in this campaign and in me: I want you to know that nothing has made me prouder than to be your champion.
Translated, the part behind the colon could read:
Ich möchte, dass ihr wisst, dass nichts mich stolzer machte, als euer Champion zu sein.
Since this is a newspaper article, it will use reported speech which requires finite verbs to change to conjunctive I wherever possible. That renders as:
Sie sagte, dass nichts sie stolzer gemacht habe, als ihr Champion zu sein.
Note that it doesn’t require a dürfen nor a habe. The full verb machen, however, changed from indicative to conjunctive.
Possibly, the newspaper decided to restructure the sentence altogether and add an additional full verb (here dürfen) to overcome any issues with pronouns. If they had left it in the way I suggested, potential ambiguity arises whose champion Hillary considers herself to be (her own?).