Do Bike stores allow you to ship a bike to them to assemble?
I think there are two questions here:

1 - Can you have a bike professionally assembled by a bike shop?  In my experience (having done this a couple of times) it will typically cost the price of their "pro tune" or equivalent - the service when they break the bike down and put it back together.  $2-300 say.  Bike shops can be a bit sniffy about this, but getting to know your local tech staff will help make things easier.

2 - Is this a good idea?  Depends why you're doing it.  If you've got a frame or components you want to put together - say you bought a particular rare steel frame that you want your gruppo from your other bike on...then it can be a great option.  If it's about buying a high-end bike for cheaper - it may be a good option but check where you're sourcing from. If it's about buying a cheaper (say sub $500) bike for less than you can pay in an LBS - I'd say a crazy false economy.  Instead, buy a second hand bike on CL or similar and save $100 or so to get it checked out by your LBS.

Slightly aligned to the point above - if I'm a bike mech, being asked to work on a cool steel frame or a high-end bike from a brand I may never have worked on is pretty cool.  Being asked to spanner a cheap bike from Walmart is going to make me feel pretty crappy about myself and question why I work in the industry.  The level of work I do is likely to be commensurate with how the work makes me feel...