Ok, thanks for that info.
It could very well be ear mites, but it could also be a bacterial infection or a yeast infection. The only way to find out for sure is to have your vet do a cytology on Monday when they open. Once the condition is properly diagnosed, your vet can give you correct medication to treat it.
Moisture that gets caught deep in the ear canal is usually the culprit for wax build up and subsequently any one of the above infections could occur. Sometimes certain dogs have a predisposition to infections due to the way their ear canal is designed - such as narrow, deep canals or heavy ear flaps. Your other dog may not get an infection, but be aware that mites are contagious so wash your hands thoroughly after handling the affected pet's ears.
In the meantime - avoid using vinegar to clean the ear. You can use just plain cotton balls for now until you can get him into the vet for the proper medications. Wipe out whatever black gunk you can see in the canal. Avoid going too deep into the canal; only clean what you can see. Do not use water or vinegar as you want to keep the canal as dry as possible. Your vet will probably prescribe some type of ointment-like medication depending on what the problem is. They might also give you a cleaning solution (you may have to ask for it) that is designed just for ears; it is made of chemicals that are drying agents and prevent the moisture from building up in the ear. You can use this solution to clean the ears regularly to avoid issues in the future. Avoid buying the products in the pet store as they are usually cheaply made and ineffective. If your vet does not prescribe any cleaner in particular, I recommend Epi-Otic by Virbac or Sebozole by Vet Solutions. You can buy either of these online, just google the names for a list of places that sell it.
If there are some abrasions on the outside of the ear (on the flap) you can put a little triple antibiotic ointment there to soothe the scratches, but avoid putting any in the ear canal until you see your vet.