What mouse? Most don't have an rgb led. Pictures and links would help.
It is very unlikely that there is a connector, but not impossible, especially if at the outer extremes of how far an LED's leads will reach.
Why have you not even opened the mouse to see what there is before this topic? I mean that would have gotten you one step closer, and generally, taking the mouse casing screws out to pop the shell off is a relatively low labor, low risk, a quick 'n easy thing to do... once you cut a slit in the stickers if any are covering the screw hole(s). Once it is open, pictures of that would be most helpful to you.
Odds are that it is either sitting flush, or on long leads, soldered to the PCB.
If it is on long leads, you have a fair chance of easily clipping the leads off at the PCB with nail clippers if not proper tool(s).
If it is sitting flush on the PCB, "break the LED bulbs" is potentially damaging either from ripping the pads it connects to, or even shorting the LED anode and cathode together if you don't check for that before applying power. You can have a reasonable check made visually, but the best is using a multimeter.
If it end up shorting leads from destructive removal, odds are it would do no more harm than to the LED driving subcircuit of the mouse, but i would not assume it on someone else's property.
The obvious solution to me, having a soldering iron... would be use it. Desolder the LED. If you lack a soldering iron, how about a pair of pliers to hold a large nail heated on a stove burner? I am serious, it seems crude using a nail but using standard soldering principles and considering it's only an LED, seems quite do-able.
If it is going to be a hardship w/o a mouse, you might go ahead and get a replacement... even some cheap piece of junk beats nothing, or just use paint or a sharpie marker to block the light? Depends on the design, again pics might help.