The vendor has done their part. Someone on your side has not signed, or has not opened a port, or has not returned a date. You have asked the person who owns it. They have not moved. You now need someone two levels up to pull a lever, not to hear the story.
A saga is how you get a meeting. A lever is how you get a yes.
One lever, one cost, one default
"I need you to tell legal to return the vendor redline by Thursday 16:00 UTC, or to tell me we are walking away. The vendor will release our slot on Friday 09:00 UTC. I recommend you push legal. If I have not heard by Wednesday 14:00 UTC I will tell the vendor we are walking away, in those words."
That is a decision they can make on a phone. The date is honest because it is the vendor's date, not a nervous one you invented.
Do not name eight people who disappointed you. Name the bottleneck and the person who can move it. Skip-levels cannot referee a team they do not manage from a narrative.
Keep the vendor on their own channel
They may not have an account. Write to them separately. They get email and a page. They can reply from the link. Do not put them on the skip-level note, and do not put the skip-level on the vendor note. Each reader needs a sentence they can act on, not a play they were cast in.
If the skip-level has an account, you need an accepted contact. If they do not, they get email too. You do not need a shared exec Slack.
You approve both packages. An assistant can draft. It should not send a threat you will not keep. The default in the first paragraph is a sentence you must be willing to say out loud.
What this is not
It is not a performance review of the person who is stuck. If that is the real conversation, it is a different note, to a different person, and it does not belong in a vendor unblock.
Use names the skip-level already knows. A private nickname for the vendor or for the bottleneck does not belong here.
If the week is already in a session, connect your assistant and cut it to the lever, the Friday hour, and the sentence you will send if they stay silent.