As long as there are no In World or in In Campaign reasons to forbid it.
There are nor rules in the PHB, DMG. or any other Rulebook that reserves any core class for a special gender.
This didn`t hold always true for special variants of the corec classes or PrC.
Some have gender restricted entrances.
Like the guardians of the blue well, 100 female and virgin Order warriors(read paladins more or less).
Or the amazon, soldiers and witches of vanga, only women are accepted into these groups.
Men in these cultures ar at best secon class citicens.
But even these didn`t prohibit the wild men(Barbarians) to fight.
on the same world, the man of gorgan, another Continent, are doing the fighting stuff almost exclusive, but female warriors(not amazons) aren`t unknown, a rarity but definetly not unknown.
It depends mostly by paladins, how he considering their upbringing.
Is it a calling or is training needed.
IMC trainig is useful but not necessary, if you don`t have the calling, all training, all Ethics all sacrifice didn`t make you one.
If you`ve the calling, and accept it, then you are one.
End of dsiscussion.
´Trainign will be very< helpful to get your powers effective, to learn what you could do with them, teach you the finer problems of ethics, and very important get you known.
Known under other Pallys, in the churchs, etc pp.
Only a paladin can train a Paladin.
