Vow of Poverty is an exalted supernatural feat, and exalted feats are explicitly stated in Book of Exalted Deeds to be aquired as gifts from powerful agents of good like deities or celestials.
If the Vow of Poverty feat in question was a gift from a deity, would it be affected by an antimagic field?
Vow of Poverty is an exalted feat.
[Exalted] feats are supernatural in nature (rather than being extraordinary abilities, as most feats are)
(Book of Exalted Deeds page 39)
Supernatural abilities are magical and go away in an antimagic field
(SRD > Special Abilities Index > Special Abilities > Supernatural Abilities)
Thus, in an antimagic field, you no longer have any benefits derived from any exalted feat. Vow of Poverty is an exalted feat, and the bonus feats it grants are derived from it. Thus, in an antimagic field, you no longer have access to Vow of Poverty, which means you no longer have access to the ability that granted you those bonus feats, which means you no longer have those bonus feats.
Note that this is stronger, even, than merely losing the prerequisites for a feat: in that situation, you still have but cannot use the feat. In this situation, you no longer have the feat wholesale. This distinction is important if you have other feats that require the feat in question: if you still have it, it still meets the prerequisites even if you cannot use it. If you do not have it at all, you no longer meet any prerequisites for it, which means you cannot use those feats at all.
And since I know you are concerned about salient divine abilities, it is true that
A salient divine ability functions normally within an antimagic field, and is never subject to spell resistance.
(SRD > Divine Abilities & Feats Index > Divine Abilities & Feats > Salient Divine Abilities)
However, not everything that a deity does is a salient divine ability. The only salient divine abilities are those listed here; the ability to grant exalted feats is not among them. Book of Exalted Deeds references salient divine abilities only once, stating that
Celestial channeling works in a manner similar to the Possess Mortal divine salient ability that some deities have (as detailed in Deities and Demigods).
(Book of Exalted Deeds page 23)
Celestial channeling, for reference, is like the good version of fiendish possession, and has absolutely nothing to do with exalted feats.
Book of Exalted Deeds could have chosen to define exalted feats as the result of a salient divine ability if the authors chose to do so. It could have even chosen to not say anything at all, and leave antimagic field out of it. Instead, they specifically went out of their way to define exalted feats as supernatural; the only significance of that rule is that it suffers a vulnerability to antimagic field. If exalted feats were expected or intended to function in an antimagic field, that rule would never have been written. The entire point of it is to have antimagic fields suppress exalted feats.
But really, being supernatural isn’t a big deal. Antimagic fields are rare, small, and easily eliminated (by killing the wizard commoner standing in the middle of it). In fact, despite writing a lengthy description of why Vow of Poverty is terrible and no one should ever take it for any reason, I have never bothered to even mention this vulnerability. It’s just extremely minor compared to all the other terrible things about the feat.
But seriously, regardless of why you are concerned about Vow of Poverty, let me reiterate that the Vow of Poverty will very, very badly reduce your character’s power, and in most cases will make your character completely incapable of keeping up with mid-to-high-level foes.
