Wow - sounds like you are getting the silent treatment from your client...but don't take it personally. Most likely their lack of communication stems from the fact that they're juggling a lot at once and are strapped for time. Could you schedule a visit to your placed candidate and drop by your contact's office while you are there? Or drop off a handwritten note? Perhaps take this as an opportunity to expand your relationships within the company at all levels and reach out to the direct manager or other Human Resource colleagues.
Situations like these remind us to set communication standards when you do have your client's ear:
1. Get a structure in place for clear communication.
Agree on turnaround times and communication methods with your client. Ask if phone calls are better than emails. Maybe face-to-face meetings are more your clients' style. Don't assume that what works for you will work for them. If they have days they know are more hectic or they know they'll be unavailable, make note of those, too.
2. Don't end a meeting without scheduling your next meeting
It's kind of like getting a second date with someone you like: If you like someone, you shouldn't hesitate to ask him or her out again. Go after your clients in a similar way. If you leave the schedule open-ended, you've created another step you have to take to set it up later (and more messages for which you'll have to wait for responses).
3. Create a "if we don't hear from you" plan
Proactively ask your clients things like "Who else can provide feedback, just in case we don't hear from you?" It creates a backup plan and introduces backup contacts.
It's not easy to keep clients engaged when they're seemingly tuning you out, but these simple strategies can help you stay in touch and prevent your client from "ghosting" you in the future.