Can you pull a credit report on your adult child, spouse - or even any other adult for that matter? If you have access to things like national ID (Social Security number), can answer security questions related to addresses they have previously lived and outstanding major debts they may have (house loans, car loans, etc), then sure you physically can.
Now, is it legal to do so? According to Experian (a credit bureau) obtaining someone's credit report without their permission is considered identity theft and fraud, even if you do not use it for some nefarious purpose. Even if you are a parent, it is a violation of federal and state laws and is punishable by fines and imprisonment.
Can they find out? Absolutely! Answering a question wrong (according to Visa), or in some other way tripping a security alert (such as if the person has intentionally put a "freeze" on their account and opted-in to some identity theft protection scheme, such as one requiring a special pin), will cause your attempt to fail and often require calling the company directly to provide additional verification information (which of course you may not have and is less predictable). Depending on the monitoring scheme, a monitoring service might even send an email or letter to a person directly if such an attempt is made (but this does vary by service and bureau).
Even if you do it correctly, all requests are logged and often are noted on credit reports regarding past inquiries - so if in the future they run their credit report they may see the inquiry they will know they didn't request (that's weird, this is the first time I've ever checked my credit...). If the person is using a mailing address you don't know about, like a PO Box or a friend's house to get mail (so you don't know about their card/statements/loans), then you will almost certainly lock the account and a fraud alert will be sent in a way you can't intercept. That's going to be an unpleasant set of conversations!
The alternative: As Mint.com suggests, just get their permission! You can request it so you can go over the importance of good credit with them, or you can tell them outright you are concerned that they may be getting themselves into debt. You can bribe them with getting them a for-fee credit score and analysis at your expense, or extort them by demanding that if they don't like it they can live somewhere else...how you play it is up to you.
But horror stories are easily avoided, and you can use the opportunity to actually get a credit report and sit down and go over it together. Do they know what a report is and what's on it? Most adults twice their age sure don't!
Parenting tip bonus: if you are open and honest and explain why you'd like to get a credit report and sit down with them to discuss it and they freak out - bingo, you've got an answer that something is wrong and it didn't cost you a dime and you didn't break the law.
Now, to be completely clear, if they don't respond well to the credit report that isn't evidence of some kind of guilt or impropriety (a thanks to the comments for pointing out this potential misinterpretation). However, what it does indicate is your relationship may be out of whack - imbalanced or unhealthy in some way. You could be in a co-dependent situation where you are covering for them so they don't have to be responsible or independent (bad for both of you), or they may be hiding from reality, or you may have have developed serious trust and boundary issues that need to be addressed, or any of dozens of other possibilities!
All of these are possible - they might be really screwing up, or maybe you are, or maybe you both are, or maybe it's all ok and you can relax. I hope this makes clear the importance of figuring out what reality is and dealing with that, which is the foundation of personal financial health and healthy personal relationships, too!