Can I use 26 1 3/8 tire to replace a 27 1 1/4? As in title, can I use  26 1 3/8 tire to replace a 27 1 1/4?

Is this workable with some elbow grease?  Any negative consequences of trying?  It is an older road bike with the original tires (27 1 1/4).  I went to a local store and closest match was 26 1 3/8.
No.  Bicycle tire sizes are a dogs breakfast of standards. The only standard that is consistent is ISO. 

According to Sheldon Brown, 27" is ISO630 and 26"  (the common sized MTB tire) is ISO 559.  

ISO630 (27x1/14) are available with a search on the internet e.g. Sheldon Brown / Harris Cyclery (not a recommendation)