When we see lost of function of the back limbs it can be secondary to severe arthritis of the hips, hip dislocation or even a pinch nerve along the back (intervertebral disc disease). In this case, we also have to consider systemic diseases as he is vomiting and has diarrhea. Other possible causes could be infections, tumors, thromboembolic episodes, etc.
In this case, I would strongly recommend to consider having him checked by ER veterinarian. The sooner we are able to treat acute onset of him limb paralysis the better the prognosis. For example, if this is related to a pinch nerve as a result of a bulging disc, the longer he goes untreated the more damage there is to the spinal cord and less likely to regain function of the back legs. If you do not have access to ER veterinarian this is what you can do until he is seen by your family veterinarian:

1- Start him on Pepcid AC (famotidine) 20mg every 12 hours. You can find Pepcid AC over the counter in any pharmacy. If you do not have pepcid ac leave without anything. Do not give pepto bismol.

2- No food or water for the next 6-8 hours. Need to give belly time to rest.

3- After 6-8 hours offer a small amount of water. Wait half hour, if no vomiting, offer small amount of rice (50%) and boiled boneless chicken or ground beef (50%) 4-6 times a day. Keep on this diet for several days then wean into regular diet.

4- Follow up with your family veterinarian.

Hope the above helps, if you have additional questions I will be glad to answer them.

Dr. Peter