I’d be amazed if a disposable, non-rechargeable item like that didn’t use easily sourced batteries. However, getting to them might be a problem. For myself, I’d consider the pump already broken and would feel free to pry, hammer, bite, whatever it takes to get inside for a peek. At worse, you still have a broken pump.
Hmmm. I looked up roundup 1 gal. powered pump sprayer (using google), but I’m not able to state with any confidence that I actually know what you have. Can you provide a model number or something?