Is it possible to trigger a reader from a standard RFID Access Control System?
Yes, RFID is just radio waves on a certain frequency. Nothing stops somebody to put a powerful transmitter on that frequency and trigger a lot of readers.
More specifically the badge system commonly used in companies for allowing entry.
Not in this case. Badges usually (almost always, but there are exceptions) have to "talk" to the reader, because the NFC tag have encryption. So even if your powerful transmitter can trigger the reader from 50m away, your receiver will not be able to receive the response. You don't just scream your ID and enter. There's an encripted talk between the badge and the reader.
This case is like putting a wifi antenna on a Pringles can and being able to receive signals from a lot of networks, but not being able to connect to the faint signals: their transmitter are string enough to reach you, but your transmitter aren't strong enough to reach them.