Heat sink installation in pcb production
Disclaimer:  I'm an Electrician, not an engineer, and I've only ever designed one board yet, so this is the product of general knowledge and the time I've spent lurking around these engineers.  A lot of product design and part sourcing works the same in other industries, but I'd welcome correction, addition or input from the engineers.
Don't underestimate the value of weighing all your options.
I don't know if it's standard practice, but assembly shops will do anything you want, if you pay them enough, which in some cases means in sufficient quantity.
The most extreme example is &quot;turnkey manufacturing&quot; where all you need is an idea and a lot of money.  These firms will consult out the engineering, development and certification and streamline manufacture of the product.  You're kind of in a position where they have you over a barrel because without the professionals they offer (including professionals to organize professionals), your idea is worth nothing.
There are also assembly services available other than shops that manufacture and assemble circuitboards.  For the shops involved in direct production, there may be MOQ (Minimum Order Quantities) for anything special you want done, and some parts are much easier to integrate.  A heatsink designed for a standard pick and place machine may be no big deal, but a heatsink that must be attached by a human may not fit in well with the assembly line structure of their shop.  They might not want to temp hire a lower wage employee to do nothing but attach nut and screw TO-220 heatsinks, especially given their shop may not be set up for the amount of space that would be needed or other costs like placing a guarantee on the result.  A huge portion of the technology sector is set up for &quot;lean manufacturing&quot;, which doesn't give them a lot of flexibility with inventory volume.  From the pick and place heatsinks I see, it's not clear if you'd need a special machine to apply thermal adhesive before placement, so not every shop may be capable.  If you contact any of them, they'll likely be happy to let you know their capabilities.
If you have a design and you want a completely manufactured one-off as proof of concept to convince investors, you might be more interested in prototyping services.  Again it's expensive and you pay a premium, but these shops have the specialized equipment to produce a high quality one-off comparable to final project without all the retooling costs associated with traditional methods like injection molding and mass panelization.
Finally, the category you appear to be in is where you're purely concerned about assembly, so it may be worth asking the board shops for quotes with and without heatsink, anything you can't reasonably get them to do just gets passed on to a later assembly stage, which may mean doing it in house.  For a significant volume product
If you're looking at quantities in high hundred thousands or millions, you're likely eligible for a lot of manufacturer cooperation, but if it's only in the thousands, and I'm guessing it is because you don't appear to have an engineer on hand, the cost of setting up for the assembly can amount to an expensive premium, and it's probably worth evaluating doing it in house against your quotes.
If you are doing assembly in house, try to reduce the assembly as much as possible to a fast, idiot proof task.  Use jigs to hold and align things, make parts and tools accessible without clutter and evaluate different methods.  Custom make your own tools to simplify doing all repetitive tasks, like a thermal compound spreader cut down to exactly the right width with bumps on the edges to control spread depth in one swipe.  If you're making a product you intend to service and want to be able to disassemble or you're using a large multiscrew heat sink where torque matters, screws and nuts may be best.  If it's a one hole TO-220 heatsink and the board is to be potted, maybe you'd rather save 20 seconds per heatsink with a pop riveter.  If you're willing to give up any hope of separating the parts without damage, perhaps thermal adhesive would be best.  Another example of a jig for thermal adhesive with a TO-220 is a simple clothes pin.  Dab of adhesive, place heatsink, pinch with clothespin, wait 15 minutes.