Good idea to follow Sandy's advice.
You will start having **MAJOR** feeding problems if you give them one thing and if they don't eat it, give them another, so prepare the bowls at the start and do not take them up to add this or that in after the fact. If they don't eat their food, remove it and that's that til the next meal, NO EXCEPTIONS.
You do need to get your dogs on something besides tinned tripe though. Most tripe is a supplement to a diet not something to be fed as a daily meal. What was the breeder feeding? Did she not have them on kibble when they were homed to you? What does she recommend now? If so I'd put them back on that kibble and then transistion to the new one if you have any more problems. I rarely feed only kibble though -- I almost always mix other things in when I do give kibble. But soft food only is going to cause them teeth and gum problems.
General feeding advice is this: Leave down 10 minutes. If they don;t eat it then lift without comment and put away til next feeding. NO TREATS IN BETWEEN. Do NOT turn mealtimes into a social interaction session where their fussiness is REWARDED by your extra attention -- eg with you fussing over them and adding things to their bowls when they stall on eating and praising them, talking to them, anything at all. Just make up the bowls, food down, 10 minutes, food removed. Meals are an easy manipulation point for dogs and they can learn very quickly that all sorts of rewards come to them by refusing to at. Frustrated owners immediately shower the dog with attention rather than ignoring the behaviour and setting strict and clear guidelines:eat or go hungry til your decide to eat and stop faffing around. Only the tiniest percentage of dogs truly dislike anything so much that they will not eat it at all! Isn't it curious that they will refuse something in their food bowl and then eat eat trash off the street, or anything that falls on the floor in the house in a nanosecond (including the shunned kibble...)? That's because they quickly learn the tradeoff to being stubborn at meals is quite a handy way to get lots and lots of attention. In the same way, many of us know our dogs start to bark and fuss as soon as we are on the the phone -- they quickly learn when we hold that thing to our head we will ALWYS give them attention if they bark, to try and get them to stop! They are smart little things!
If you are feeding more than one dog it might be adviseable to feed in their crates if you start to have any issues with them fighting, guarding, or one trying to eat the others' food.
The throwing up could be due to many things and may well have nothing to do with changing food. Puppies (and dogs) sometimes just vomit. I have never had dogs vomit up food after changing but some may well do that. I feed my dogs different things almost every day so personally would side with your husband that just changing over usually isn't an issue. I never did the slow changeover but at the same time this sometimes does work better with some dogs especially puppies. BTW if your dog continues to vomit for more than 24-36 hours, be sure to get him to a vet.