Yes a mirrorless camera can do a great job at Macro. Macro photography on an inter-changable lens camera does not depend on a mirror (or lack of a mirror). You need a macro lens. This can either be a lens you "make" like nikono mentioned or a dedicated macro lens. I've tried most of the various techniques and a dedicated macro lens will give the best results.
If you want a macro lens, search for an E-Mount macro lens. There are several good options. Sony makes a 90mm macro lens Zeiss makes a 50mm. Both cost around $1000. Between the two I would highly prefer the 90mm. The reason is something called minimum working distance.
Minimum working distance is the distance from the front of the lens to the subject. With something...