Will 26 x 2.0 tire fit on bicycle with 26 x 2.125 size tire
Yes. It will work fine. 

Judging by the description of the rims, it is likely they are Weinmann 519 or Zach 19s, or something very similar. Most of these generic double wall rims fall right around 19mm for inner width. This means they will comfortably hold a tire from 1.20" to 2.50". Many if not most big box store bikes use this size for that exact reason. There are a lot of readily available tire options and they are one of the more mass produced sizes so the cost of purchasing the rims as well as tires is cheap.