In my experience, its not unusual for assembly houses to not give prices of individual components. This is because that is where their profit is. They tend to be able to arrange for better pricing with distribution because they can buy more than one company itself can. That is also why they want you to use parts they already have.
In one case I was involved in, the assembly house was involved early in the design when a component was highly priced due to it being so new. Over the years the price dropped dramatically (like $50 a pop) but purchasing never new because they didn't purchase the part directly. So the assembly house just pocketed the difference and never told us. The only reason we found out was that the account manager let it slip that the price had dropped significantly (and thus we should use it in a new design) and this set off alarm bells that lead to the discovery.
In my experience, assembly houses that keep stock are more than willing to give you a list of their components. If they don't have to hold more inventory and all that expense they are usually more than willing to do it that way. So this is unusual.
I would advise you to get quotes from a few sources. The assembly house will want to tell you that they should be designed up front in the process and blah, blah, blah. You are much better off to pay an expert to do DFM type of stuff since you'll pay a lot but one time. If the board house is involved with this you typically end up with a per-unit cost. They won't call it that but they have to pay for their time some way and that is their common billing method.
Then treat the assembly house like a service and look for a good deal on a per unit basis. You'll probably want to make sure you have a contract that states what will be delivered and who pays for rework and parts that fail testing and even who does testing. There are lots of variables in the cost equation: money, time, testing, quality, responsiveness, integrity, flexibility, ....
After going through so much I can see why some companies just say, "Screw it" and get their own assembly equipment. But my company isn't that large (yet) so we have to make due.
Cheers.