I don't use it much but the wife is a fiend at it.
There are different systems. American FaSoLa as used by Sacred Harp groups is not the same as the classical European version.
At any rate, she sez that in the classical system, you make accidentals by changing the vowel.
Change to 'i' as in 'tea' to raise one half step.
Do > Di
Re > Ri
Mi > Fa (think about it)
Fa > Fi
So > Si (careful with this because Ti is Si in some systems)
La > Li
Ti > Do
Change to 'e' as in 'hey' to lower one half step.
Mi > Me
So > Se
La > Le
Ti > Te
Of course you don't lower Do or Fa by a half step.
And it doesn't work with Re, but given how accidentals work in real keys I don't think you'd miss it.
Hope that helps
-Glenn