It's a little unclear and colloquial. In this case, "to improve" means "to make improvements" !
You could write it more clearly as:
Retailers are hard-pressed to IMPROVE *THINGS*/*ANYTHING* when they have no idea what is wrong.
Retailers are hard-pressed to MAKE IMPROVEMENTS when they have no idea what is wrong.
hard-pressed in sentence means "in a difficult situation", it means, it's difficult for them to make changes if they do not know what is wrong.
"hard-pressed to __(do something)__" is rather colloquial (口語的)
I might write this as:
Customers can help to improve THEIR future shopping experience by filling complaints (directly) to the retailer, instead of complaining to the rest of the world. It's hard for business to improve for their customers when they don't know what's wrong.