Those of us who remember the Strategic Arms Limitation Treaties of the 1980s and 90s are familiar with the phrase, "Trust but verify." It's a great lesson for leaders.
Trust but verify was the maxim under which we inspected nuclear weapon destruction sites to ensure the weapons were actually scrapped. It's not that we don't trust you to uphold your end of the deal, we said. It's just that the stakes are high enough that we owe it to the world to make sure -- it's just due diligence.
That applies to leadership this way: You have to trust other people to do their jobs, and you do. When another department makes a commitment, or the process requires you to depend on them, you trust that they'll come through.
When you delegate to a team member, you have to trust them to do the work. If you don't, you shouldn't delegate. It's dangerous to communicate a lack of trust; it damages your relationship with that person, and it undermines his or her confidence.
But remember, you can delegate the work but not the responsibility. When it's important, it's your job to make sure it gets done. To guard against mistakes, misunderstandings, or failures by vendors or outside sources, you have to check. Go ask, go look, do something to make sure all the materials came in, all the phone calls got made, so-and-so signed the document, whatever. Especially when you see a potential choke-point, someplace where one thing can stop the whole works, you need to check.
My suggestion: Establish a routine that puts you in the area asking questions periodically. That way when you need to check on something, it's just another question. Or maybe it's just looking. Either way, it's no big deal.