My Mom has Alzheimer's and she started to vomit everything she eats. Her temperate us 87.7 - Any suggestion on what it could be?

Hypothermia is a medical emergency that occurs when your body loses heat faster than it can produce heat, causing a dangerously low body temperature. Normal body temperature is around 98.6 F. Hypothermia occurs as your body temperature falls below 95 F.
When your body temperature drops, your heart, nervous system and other organs can't work normally. Left untreated, hypothermia can lead to complete failure of your heart and respiratory system and eventually to death.

Please call 911 immediately

Either get her to ER yourself you call 911 RIGHT NOW.

Emergency room visit now. She may have aspiration pneumonia from choking and swallow deficit. The very subnormal temp may be equipment failure. The swallow deficit needs to be evaluated immediately and she may need xray of lungs for pneumonia. A swallow deficit could also be from an undiagnosed stroke. In any case, this is an EMERGENCY evaluation now.

It sounds like it could be Gastroesophageal reflux disease (GERD). I've known several people with GERD and depending on it's severity it can range from heartburn on up to not being able to keep food down. Your mother's low body temperature is also concerning, though not sure what that would be.

Please contact her doctor for information. She needs professional help.