If it is private property, then functionally,"yes". Within reason, that is exactly what they should be able to do. If you do not wish to abide by the rules that they have set for their private business, then you have the option to take your business some place else. At that point, nobody's rights have been infringed and every one, except the busy body nanny staters should be happy.
In MO, cities/counties etc have the option of banning open carry outright-many do, many don’t-so open carry is a mish-mash of yes/no and you, the citizen have to know which jurisdiction you are in and what it’s rules are. All completely w/in the MO code.
Concealed carry is permitted w/endsorsement (excepting certain premises-courts etc) unless restricted by the property owner/manager via an 11x14 sign prominently posted with black letters (although the red cirlce over a firearms is probably enforceable as well).
Violating such signage is not a crime, rather a citation if LEOs are called and you have refuesd to leave (with caveats).
(If carrying concealed, how would anyone know?; but of course honor other’s rights and the law; let them know and shop elsewhere!)
For details, go to mo.gov and search for “concealed weapons”.
MO CCW trainer/NRA R&P Instructor etc..
“Within reason, that is exactly what they should be able to do.”
Opinions vary as to what is “within reason” and why or why not. That’s usually where the major problems lie in these situations.