As stated above, it does depend on the state, but let me pose the situation in another way. Suppose it was a private home, an the owner left the door unlocked at night when he went to bed. Can I walk into the house and only be considered trespassing? The homeowner's failure to lock the door is not legal justification for entry, as we have a right to privacy the courts have consistently upheld.
When we talk in a legal sense about private property, and especially in the case of a business where property is maintained and a target for thieves, "breaking" the plane of the doorway outside of business hours would most likely be considered breaking and entering as well, and/or burglary. If someone is inside, then most likely robbery would be the charge.
This depends to a large extent on the rules of the state in which the offense was committed.
In some states, the information that you have provided would not be enough to answer the question. It is often the case that breaking and entering (also called burglary) is a kind of trespassing. You are trespassing if you go into some place without permission. You are breaking and entering if you go in with the intent to commit some other crime while you're in there.
In such a case, I'd call this burglary unless the person can show some reason why they were going into a closed store without intent to commit a crime.
It is also the case that trespass is typically seen as a tort rather than a crime.
This is almost like asking the question of if no one is the forest and tree falls down and no one was there to see it does the tree still fall. Of course it does. So yes this is exactly trespassing. Sometimes the level of getting in trouble depends on how trespassing in places like that is enforced.