If she’s drinking 40 ounces per day, I think you’re right that she needs some solids. Unless she’s a comfort eater. Does she seem to be pain free and happy most of the time?
Most babies under six months will push food out because they have not completely lost that reflex to push things out of their mouths with their tongues. This is a protective reflex meant to prevent choking. It takes practice for them to learn to move the food to the back of their mouths and swallow. So, keep trying and I’d try different consistencies. Most babies I think do best with thinner consistencies like Amy said, but Ellie did best with suprisingly thick cereal.
I’m not sure about the vomitting, though. Is it because she gags? Or do you think she’s not tolerating the cereal?