Can I ask my contacts for opportunities?
You can ask, but if you have a very strong relationship with your clients you won't need to - they'll volunteer the information without prompting or hinting.
Is it the norm?
To ask outright? That's a judgment call on your part. I wouldn't do it. Others would.
Or is it likely that I will be exposed and get in trouble with my current job?
That would be my guess. If you're lucky you'll merely get in trouble. If you're unlucky, you'll be unemployed and the client won't want you either.
If I say "confidential", does it bear any weight?
Not really. If you feel you have to explicitly ask your customers to keep quiet, then I'd say you don't know them well enough to trust them this much.
If you're serious about working for one of your clients, the trick is to be totally amazing and on top of things and an obviously fantastic fit for their organization. You don't even hint that you're looking to work for them. If they decide they want you bad enough, they'll approach you (which puts you in a great position to negotiate salary).