Given that there's already a half-sexed boer'alvar valkyrie (meaning penis and no vagina) in the game, it would be pretty clear that It Doesn't Work That Way even if the codex entry didn't also say that it... doesn't work that way.I'd personally like to throw out there the idea of binding to Lumia can come with perma-vag, whether you always have one that replaces your cock or it grows underneath your cock. Just would like to embrace the full extents of the divinity of the sun goddess.
Sexes: All valkyries appear as women, though they can have any genital arrangement.
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Much like kitsune and the Astrida to their respective gods, valkyries are the foremost extension of Lumia's will in the world. They strongly resist any effort to change their physical being against their will, and have a measure of resistance to mental change as well. Despite these similarities, valkyries are the least static of any godsworn. They are perfectly capable of changing minor details about themselves using alchemy or other magic.
Unless you're blindly experimenting with TF items, it's really hard to mess up the Champ's genitalia by accident because the alchemy system allows you to refine almost all TF items so they only target specific body parts. Don't want to alter your genitals, avoid using anything refined with Lush Orchid or Turgid Mushroom.
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