So there are normally two major concerns that a shop has when a customer supplies parts
1) as previously noted, liability and warranty. I'm not guaranteeing or dealing with warranty concerns when I don't have an invoice with a PO number on it that I issued from a vendor. Regardless of the quality of the part. If a customer or client is willing to sign their life away then that's fine.
2) lead time. Which normally is a different scenario than currently due to supply challenges. Oftentimes customers will find something a few dollars cheaper and want to order it in which on the initial endeavor prior to work on the vehicle starting, isn't of major concern. But when I have your car up on a lift and something additional is needed and I can have it same day or next day and the customer wants to save $6 to get it in 3 to 4 days, I'm making them pay labor for the difference in delivery time in full. If they want to pay an additional 24 hours of labor for the 3 days that that lift is sitting occupied in order to save $6 on a part, that's their prerogative.
(I know that this isn't relevant to your current example or situation, but in a normal supply chain it's a slippery slope where people start looking oh I saved this much on this part I can save this much on this one I want to order this one in as well. And then they also get this sense of arrogance that they can source parts cheaper than the shop and project this idea that they should be working in parts procurement for them. It's a bad look in a normal supply chain world.)
With regards to the parts markup. It depends upon the cost of the part. There are some things that I only mark up 3 to 4%. For example on EFR turbo, if it's a $1,200 part I'm only marking it up 7% because #1 there's MAP pricing, #2 a 7% markup It's less than a hundred bucks, I can explain to you if I buy it and if there's an issue or defect and the invoice is in my shop's name, that I can contact Borg Warner or fullrace explain the defect or what occurred and they will issue a warranty without question knowing the chain of ownership and quality of installation and engine management that was being used and I'll go over the parameters with regards to tuning what wide band was done how we were setting up and measuring the speed density etc when the unit failed. There won't be issues with regards to getting a warranty replacement, that $100 is peace of mind. Whereas if somebody brings in a unit off the street, And I install it and I give the customer all that same information and they take it to the supplier or vendor where they purchase it there's still a difference in chain of ownership and all I can say is it was brought to me and I did this. And I'm only telling the customer that I'm not dealing with the vendor that you purchased it from unless you want to pay labor rate for that communication, which is also going to exceed more than that $100.
On larger stuff such as supercharger kits that can be over 3 grand, That is where sometimes the markup will only be two or three percent for the same reason. Now a little miscellaneous plastic clips and on fluids that I buy in bulk, my markup might be over 80%. I use a parts pricing matrix that is based upon what my cost is if it's a high dollar item my markup is far less than something that costs less than $2. If it's a low-cost item I'm marking it up but I buy them in bulk at a volume where my discount is so significant that you likely cannot even order it from Amazon for what I'm charging you after my markup. And if you can you probably spend 30 to 45 minutes hunting for it and confirming that it actually is the size and fit that you need in order to save 45 cents per a piece. If you decide that your time is worth $0.90 an hour that's on you I'm not here to make that decision. (And again when I say you I mean a customer as a whole, not you personally). I'm not savings of $0.45 per clip is also subject to that same if it takes 2 days to deliver and I literally have it here, that's 16 hours of labor that I'm charging based on the opportunity cost of that vehicle being in my bay.
My general rule is I'll explain the situations and scenarios to me obtaining the parts to a client before they bring their vehicle in and once the vehicle is here, they're not getting any more parts for it themselves unless I specifically say, I've gone through my standard channels, I've searched out into the ether of the internet, and I can't find it from a source that I trust. Here is my best ETA. If you can beat that ETA with a tracking number for delivery, let me know where you would be obtaining it from, send me a direct link. Because if they're able to do that I'll still want to purchase the part for warranty purposes, but I will sell it to the customer at cost with whatever warranty the manufacturer provides on it, because that's the rare instance that I can't get something but the customer has been able to show me where I would be able to get it. Which sounds like your situation. That's mutually beneficial, but I still want to take direct possession of it from the vendor as purchasing it for warranty purposes I'm just not going to mark it up for you.