Dr. Fiona's computer is still giving her problems, so she asked me to answer your questions.
The timing of his vomiting, along with the fact that he is happy and feeling good, makes me think that something within his stomach may be keeping it from emptying properly. I do want to confirm that it is actually vomiting that he is doing, however, and not regurgitating. Vomiting is an active process, preceded by retching which begins in the abdomen and then ends with the food being actively propelled from the stomach. If you have ever had the dry heaves yourself, you know how forceful this can be. Regurgitation, on the other hand, is a passive process. A dog generally lowers the head and then food or liquid is coughed out or just runs out passively. Although there may be mucus or liquid mixed it, it is not digested and may smell very foul. Regurgitation can indicate a serious swallowing disorder, where food and water accumulate in the esophagus and don't make it to the stomach at all. This does not really sound like what you are dealing with, but it is good to make sure.
If Riley is truly vomiting food from his stomach, that soon after eating, and there has been no diet change except for the type of rawhides, then I would blame the rawhides before I worry about more serious things. A lot of dogs have trouble with any type of rawhides. Even though they are digestible, it takes a while. If a dog swallows them quickly, they may sit in the stomach for hours. Often, they are vomited up in the middle of the night, when the food has already left the stomach, but if they block the stomach exit at a time when the stomach is full of food, it could keep it from emptying in the right direction. Of course, it is also possible that something else is going on. He might have chewed up and swallowed something indedible. The things dogs swallow might surprise you--anything from cat litter to carpet to Nerf balls. Any of these could cause him to vomit food shortly after eating. So could some sort of overgrowth of the stomach lining. This is more common in smaller dogs that tend to be hyperactive--in this situation, a fold of stomach lining (or redundant gastric mucosa) can cover the exit like a wad of indigestible material. Another possibility is that the food has spoiled--gotten moldy or rancid. If that is true, there may be toxins in it that make him nauseated when he eats it. If it is a brand new bag of food, the company may have changed the ingredients and there may be some new ingredient that he does not digest well.
If it really is just the new rawhides, the problem should go away within a day or two, assuming that you don't give him any more.