My 14.5 months baby has been coughing and vomits for two weeks now. They say she is throwing up sputum.?

Her airways are all blocked up and the only way she can clear them is to throw up. Have you tried taking her into the bathroom, running the shower hot and letting her breathe in the steam? This opens the airways and may give her some relief. Two weeks sounds like a long time. Don't put up with the stress, take her to the doctor again, just in case. All the best.