The red spots are likely petechiae from the pressure of vomiting, which happens to quite a few of us during morning sickness lah. They should fade on their own within a few days to a week or two since you're not having any swelling or other symptoms, so that's a good sign. Just avoid squeezing or picking at them, keep your face clean and moisturised, and they'll gradually disappear. If they're still there after a couple of weeks or if anything changes, definitely check back with your doctor or PD just to be sure, but honestly most mums find they resolve without needing to do anything special.