New cat vomiting after move, resident cat hisses. Should I worry?

Moving into a new home is stressful for a cat, especially if there is a resident cat who doesn't like him. Given the move and the presence of the new cat, I would be inclined to think Simba is throwing up from stress. If he's never had Frisky's wet food before, then that could also be a cause of the vomiting. You can try a bland diet to see if that helps with the vomiting before having to see a vet. I recommend withholding food for 12-24 hours, then feeding a bland diet of boiled white meat chicken (no bones) in small amounts several times a day for a couple of days. Then slowly add back in his normal diet. If the vomiting continues in the absence of food, he vomits the bland diet, develops diarrhea, stops drinking water, starts vomiting water, or becomes very lethargic, he will need to see a vet right away.

Given that there is another cat in the house, I would slowly introduce them to each other. Take Simba, and put him in his own room with his own food, water, and litter box. Keep the door closed, and for a few days, feed them both treats on other side of door so they associate each other's smell with a good experience. Then let the other cat in Simba's room while he is in a cat carrier. They can smell each other, but not be in contact. If either becomes upset or show signs of aggression, remove the second cat from the room, and try again later. Praise them when they interact appropriately. When they are interacting well with each other through the carrier, you can let Simba out of his room, and see how he interacts with the other cat. Praise them when they interact appropriately, and separate them with any sign of aggression or upset. I would get a cat tree with lots of shelves to give the cats vertical space to get away from each other. I would keep Simba in his room when you go out until they are comfortable with each other.