We used a latex primer that was left over from whenever. We used it on some hollow core doors and trim. The primer was left to dry and now gives off an odor much like vomit. Will the finish coat of latex paint mask this smell when it dries or do we have to take other measures? This is a first for us.
Sounds like you used some spoiled latex primer. Latex and acrylic primers can spoil and even grow mold after some time. And yes it smells really bad when this happens. You will need to do one more step before the finish painting can begin.
You need to re-prime the doors with an oil base primer. Use a fast drying stain blocking primer like Kilz Original Oil Base or Zinsser Cover stain. Both products can be found at any paint store or Home Depot and Lowes. This new coat of primer will help seal the bad primer making sure the smell doesn't bleed through the finish.
Now is a good time to check the doors for thick edges or brush marks. Sand the surface if you think it needs it and wipe the surfaces with a tack rag or damp cloth to remove any dust.
Allow the new primer to dry for a couple of hours before applying the finish paint, over night is better.
Discard the old primer. Once it is on its way back to nature the primer can't be used.