Yes an urgent care center can do blood work without the need to go to a different doctor or the emergency room.
Urgent care centers are setup to do just about anything that a doctors office can do and sometimes more.
The emergency room is equipped to do even more than an urgent care center but I've had blood work done at urgent care centers before.
Just about any of the tests that your doctors office can do for you the urgent care center can do the same exact tests and all you need to do is to walk into the urgent care center.
With a doctors office you need an appointment but if you need blood work done faster instead of waiting for an appointment then you could go to urgent care and they can do the blood work faster.
It still can take awhile to get the test results done depending on how busy the urgent care center is though so you may or may not get immediate test results from your blood work that is done at the urgent care center.