Creating new entity and converting an existing entity are not the same.
Do we need to notify/update our employees, clients and vendors about
the new EIN?
Is there a way to let the IRS know?
Know what? You file a final tax return for the LLC (marking it as final on 1065 or as business closed on schedule C), and a first year return for the C-Corp. Have your accountant file the appropriate paperwork for merging, basis allocation, etc.
What happens if we missed some?
If they send you 1099 with incorrect EIN, you'll have to kindly ask them to correct. It can create a mess with the IRS because it may confuse them as to who get paid for what and by whom, so try not to miss anyone.
I'm not a tax professional and this is not a tax advice. Free opinions on Internet forums are worth exactly what you paid for them. For a proper tax advice please talk to a EA or a CPA licensed to practice in your state.