Is the answer
As the explanation goes here:
Also a group
A rockband callded EYE (is a NY based band, pronounced the same as the single letter) exist(ed)!
Check out the reason why
Usually an introspection (for reasoning why) statrts with one's self / i
Move gently to disappear
Close your eye gently to make the things disappear
Chase after a little money
The little money is interest, usually denoted by letter 'i'.
Stores quality finds
Usually useful information is gathered and displayed with letter 'i'.
My own storage space
In Apple poducts storage space is denoted by 'i', such as iCloud!
Health club fight
To be checked ...no information...
Hold onto a grudge
Grudges are usually caused due to EGO, which is the reason for holding/keeping a grudge alive, for which the cause is 'i'.
Round and around
EYE / i can go round and around remaining the same !
Trolley's location is usually identified at 'i' center