The bulk mulch just sits in a pile that I draw from as needed — some people cover it with a tarp if they’re not going to use it soon.
As far as leaves and shredding, some people make smallish leaf piles in a convenient area (like where they might compost or store mulch) and run those over with their mower.
Whole leaves piled up on their own — not with other yard waste, but a leaves-only pile — will get all dry and crumbly in time aka “leaf mold” that makes good mulch.

Also the former owner told me that she used shredded leaves & grass cuttings for her mulch. Her husband mowed their lawn and had a bag attachment so was able to collect the shredded leaves & grass but our mower person doesn’t have a bag attachment. Other than getting my own mower, do you have any recommendations on how I can shred my leaves? Or maybe I should get a small push mower so I can produce enough for the veggie beds (we have 5 acres so there’s no way I can mow that with a push mower myself). Any advice would be great!