Try Dinamo. I'm not sure if they do just local moves. I used them to move stuff from the USA but they were dependable. If they don't do it they will know someone that does.
Best thing is to call around and get some quotes and look at the reviews. On Google you can look up Fletes y Mudanzas. I have used this company but it was a few years ago.
Fletes en Capital Federal, Gran Buenos Aires e Interior del País. Precios y Medidas de Vehículos. Empresa de mudanzas con soluciones a Medida en Fletes.
Betsy as usual is spot on. I have had friends use this company and also Fletes Ya. If you give them a list of your items and addresses they should be able to give you a quote. It's good to get multiple quotes and then many times they will match the lowest quotes.