You might it difficult to find a good shop that will let you provide the parts. I'll tell you why, based on my experience at an independent shop. Because we finally had enough and refused to do it.
Customer calls around to save a few bucks and brings me a water pump from Kragens. (Now O'Reilly's, I think) I get so far along and discover the pulley holes don't line up with the water pump hub. So now we get to play phone tag. And then who gets to explain the problem to the parts store? And gets to find out that mid-year there was a change from 40mm spacing to 45 mm spacing and surprise, surprise, nobody locally stocks the 45 mm version? Should I have checked earlier? Yes. So I have to eat the wasted time there. But do I get paid extra time to make a bunch of phone calls to help the customer get the right part and a discounted price? Nope.
Next case: blown engine, Hard to locate a used one. Found one, added markup, gave estimate. Customer needs to think this over. He calls back, boasting about what a great price he got on an engine and we really ought to try this wrecking yard. (It was the same one we called.) So engine goes in, car starts, car blows oil smoke that would make James Bond envious. And it wasn't just preservative in the cylinders. Nope. This thing was burning a quart of oil every 500 miles. The junkyard would be happy to refund the money. Just bring the engine back. The customer thought we should do the second R & R for free. My boss said, "Nope. There's nothing wrong with my installation." Had we sold that exact same motor to the guy, we would have been hunting for another one and replacing it for free. That guy sure saved himself three hundred bucks!
I can't even begin to count the number of starters and alternators that failed within weeks of installation and which I was expected to warranty. Again, the question is: Is there something wrong with the installation? Because that's all I charged for, so that's all I'm liable for. Now if I sold the guy the parts and the labor, I own it all. The customer doesn't give a rat's if the part failed or the installation was done wrong. He wants it fixed. Now. And it's not his problem if the parts store gives me grief. That's my problem.
For mods, things seldom go totally smooth. There's always a clip that's missing. A bolt that is a hair too long and hits something else. Some foam tape is needed to prevent squeaks. If the shop is responsible for the whole job, they'll deal with that stuff as it comes up just to finish the job and have you be happy. Otherwise, they'll start calling and bugging you to find whatever. Or to add to the cost estimate so they can find it. Or maybe they'll just leave that foam tape out and you're stuck with the noise. And whose fault is it? The kit you wanted installed didn't come with it, right?
Unless the shop is marking things up 100%, it's cheap insurance to let him make a little profit on the parts. No matter what happens, it's his problem while it's under warranty. If the vendor wants all the original packaging to return a defect, it's the shop's problem if they don't have it. You say, "You work it out with them. I just want it fixed." And if things really go south and you end up in small claims court, there will be no question about who is liable. Nobody can claim that you got the wrong part or that the manufacturer sent a defect.