As far as I can see, a company called Pampa Luggage had opened a service at EZE back in 2014, but they seem to have closed shortly thereafter. As far as I can tell, there is currently no "fixed" service of this type at Ezeiza International Airport. However, someone seems to have identified this and appears to be offering a comparable service, although I get the feeling that they are not actually located at EZE.
Please continue you research here....
Luggage Storage Buenos Aires | StorageBA
If you can, please come back and report your findings.
Cheers,
Alex / Gaucho100K