Is it: I am going to Meijer's. Or: I am going to Meijer?
Technically, "I'm going to Meijer" is correct. However, it is a peculiarity of American English that we occasionally refer to family-owned businesses in the possesive form; for example, anyone raised in the Detroit area will most likely say "My Dad worked at Ford's for 30 years" rather than the more appropriate "My Dad worked at Ford for 30 years."
As an aside, once upon a time the Meijer chain was known officially as "Meijer's Thrifty Acres," so referring to it as "Meijer's" was just shortening the store's name.