Will PCB assemblers solder wires to the board?
Yes, that should be no problem with most contract manufacturers.  However, the only way to know if any one manufacturer will do this is to ask.  Generally manufacturers will perform a variety of tasks, including hand soldering when necessary.  Of course they will take the total task into account when quoting the job.  Generally though, it will be cheaper to have the CM hand-solder additional parts than for you to do it after receiving boards.

Of course all this needs to be properly specified in the documentation package you give the CM to quote on.  The board drawing should show where the wires go, how long they are, how to strip them, whether to tin the ends, etc.  If the process is complicated enough, you have a separate drawing for that.  Any manufacturer not set up to do that will just not provide a quote.