Question: I'm my experience, I found its customary to give at least two weeks notice when separating from your employer. How does this work when you're in a position with access to all systems? 

There have been two other individuals in my time at my current employer that quit on the spot. When I enquired about this, a coworker informed me that this is normal because of the access those individuals had. 

I would like to leave on good terms and I'm not sure what the etiquette is in these situations.

Answer: 
It is very unlikely that those individuals gave zero notice. What almost certainly happened is that they gave the expected amount of notice, but the company decided not to require them to work that notice. That option is always open to a company, and is frequently used when the employee has access to confidential information. Typically the person still technically works for the company through the notice period, and is paid, but they don't have to show up for work and their access privileges are revoked. Alternatively the company may have agreed with the employee to waive the notice period, so the employee could start work at the new company immediately. 

in the first case it's the company's decision, not your colleagues. In the second both parties have to agree. If you are in this position, you should give the normal period of notice and wait to see if the company wants to take either of those options.
It depends on how bureaucratic the company is (regarding the access). it might take them a while to revoke the access. Also it depends on how critical your role is and what kind of mentoring you need to organize for whoever will take care of your responsibilities once you leave (or how you are going to document things for the next guy).

The fact that other people were able to do it, doesn't necessarily means that you can use them as reference, as you said, they left under a different circumstance.

Try to be honest with them without sharing too much personal details, if you want to leave due to a new/better opportunity then talk to your manager, schedule a meeting to talk about it. If your main point with this question is the access issue, then ask them if you can sign a Confidentiality Agreement.