This is a great question, and your answer is: it depends.
There is something in the law called the "Emergency Exception" (relating to emergency responders entering private property without the consent of owners when there is a verifiable emergency). It depends on your state what the exact limitations and the statues, but I'm pretty sure every state has one of these.
In addition, the facility may have some sort of Emergency Action Plan (EAP) with local emergency crews. It depends largely on where you are, and what kind of facility it is (is it large, prone to emergencies, ect), but many places will establish an EAP with first responders in case of emergencies. For example, in my area the local schools, malls, and assisted living homes have EAPs with us, so we'll know where to go in the event of an emergency, who can get us access into places, and so on. However, it's very possible (and unfortunately, probably also probable) that your facility doesn't have one of these, or is capable of getting one (not every facility is "worthy" of coordinating an EAP with local first responders, and most agencies would rather not bother).
We can still find ways of getting into the gate. I actually carry a battering ram in my bus (laugh at me all you want, call me a whacker, whatever, I'm prepared).
And btw, if there is an emergency, we probably won't be the first on scene. I'm sure somebody will be there calling 911 and holding the gate open for us when we arrive.