I think June/July could be crazy hot there
The temperature is the same year round. It's on the equator.
Anyway, Dec-Feb is best for the east, but wet in the west ranges. June-August is the best time in the west, but with a short drier spell mid Dec-mid Jan.
It depends on what you are climbing. Dec-Jan are pretty reasonable for both ranges, so it is a good time to go. If you are climbing just the western peaks, June and July are prime, but eastern peaks are very wet there.
Basicially:
June-August are best for Chimborazo, Illinizas, and Carihuairazo. Those peaks are also reasonable for mid December and January and often the weather can be good then.
December through January are best for Altar, Antisana, Sangay, and Cayambe.
Cotopaxi goes well in both seasons.