Hello, my name is***** and I have over 20 years of experience as a veterinarian. I'm sorry to hear about your pups Riley and Bogart getting ahold of corn cobs last night. This is a tough one as the pieces in the stools and vomit today may be the last pieces of or there may be more. The good news is so far they seem to feel great, which likely means there isn't a full obstruction.
I don't find giving oral vegetable or animal fat based oils very helpful as they are just digested and don't last long in the gastrointestinal tract, and they can lead to pancreatitis if too much is given.
To help I recommend starting with a Vaseline (or any generic petroleum jelly) sandwich. Take a slice of whole wheat bread, slather with vaseline as you would peanut butter on a slice of bread, cover with a second slice. Feed one sandwich to each dog piece by piece or they will make a huge vaseline mess. This adds fiber and hopefully will make any remaining corn cob pieces slip rather than catch. The Vaseline is nondigestable and should coat any pieces allowing them to just slide through.
For the next few days you can try feeding a bland diet with lots of fiber added in small meals several times a day, 4 to 6 meals is ideal. We do this so the pieces are surrounded by ingesta, and the gut isn't irritated by contracting on just sharp pieces.
A homemade diet for this is 1/3 boiled, lean hamburger or boiled, white, skinless chicken, all fats drained off the meat, and 2/3 boiled, plain, white rice. Add 2 tablespoons of canned pumpkin, (not pie filling, just pumpkin), to each meal for fiber to help push pieces through.
You'll need to check stools frequently for pieces in the next few days.
Signs that things aren't going well, and the pup needs an immediate veterinary visit, are continued vomiting, a tense painful belly, lack of an appetite, a fever (more than 103.5F rectally) and lethargy.
If you aren't seeing fragments in the next day or two a veterinary visit and an abdominal radiograph to see where the pieces are can be very helpful.
If it doubt it is always best to have them checked.
Once the pieces are passed and they seem to be feeling well then start mixing in their regular food, adding a little more regular and less bland at each meal. It should take a week to get them converted back to regular food.
Let me know if you have any questions.