Yes
According to the D&D 3.5 FAQ, page 47:
Just because you're barred from benefiting from a feat doesn't mean that you don't have it any more.
The Apostle of Peace must have the feats Vow of Nonviolence, Vow of Peace, and Vow of Poverty, all three of which specify that if you intentionally break your vow, you lose the benefit of the feat. However, you don't lose the feat itself, and, per the FAQ, you can still use them a meet prerequisites.
While the rules say that if you can't use a feat if you later lose any of its prerequisites (PHB 87), no such rule applies to prestige classes (DMG 176). The feat rules also strictly define "benefit" as that specific named subsection of a feat which determines what abilities it grants; in other words, should one lose the "benefit" of a feat, the rules are clear that this refers to part of the feat, and not the whole feat.
The feat rules also clarify that a feat being a prerequisite for another feat means to have that feat, and although no such definition is made in the rules for prestige classes, neither does it make any reference to requiring the stated benefit of a feat to be active.
Thus while it's clearly the thematic intent of the Apostle of Peace prestige class that you be in good standing with your vows, rules-as-written you don't have to.
The only problem is that it kind of sucks as a caster build compared to a regular cleric. As Matthieu M pointed out, the prestige class has a prerequisite of 10 ranks in Concentration, which generally requires 7 levels. The only advantage, if I calculate correctly, is that at precisely character level 16 you can have 9th level spells a level early. The drawbacks include that you have d4 hit dice, are forbidden from wearing armor, and have to burn a lot of feats up-front.
