I hesitate because some of those custom pcs from the big shops like Dell and HP have custom motherboards, psus and connections and can't be swapped with regular components. I'd search for that model and see if you can find someone with experience.
Most, if not all OEM PC's nowadays are ATX form factor, OP has nothing to worry about. Considering the age/date of the components inside, I'm confident OP will not have any problems with a new PSU.
I myself have upgraded a 3 year old HP computer with a new PSU and GPU before.