I live in a city with plenty of cycling shops, but they are all out of 27 x 1 1/4 tires. Each shop gives me a different answer about the possibility of using a different size of tire. Can I fit a different size onto an old 27 x 1 1/4 rim?
You need 27 x (some number of inches given as a fraction). So 27x 1 1/8 would fit, 27 x 1 1/4 would fit, etc. Other sizes won't fit. As dlu's answer points out, an easy way to get this right is to ask for ISO 630 tires.
That being said, 27 x 1 1/4 tires are still made in plenty by brands like Continental, Bontrager, etc. so your bike shop should easily be able to order one for you in a few days, or you can buy them online.
See Sheldon Brown's page for details on how tire sizing for bicycles is a mess.
"Brown's Law Of Tire Sizing:
If two tires are marked with sizes that are mathematically equal, but one is expressed as a decimal and the other as a fraction, these two tires will not be interchangeable."
Sheldon has another interesting note: Very old (60's) 27" rims had straight sides versus modern hooked sides, so they can't run as high pressure. However, he says if you get a tire which says hook edge rims only, you can put it on an older straight rim, just not at as high of a pressure.
Finally, tubes for 27 x 1 1/4 tires are a different matter. You can use 700c tubes in 27 x 1 1/4 tires (The right size should be 700 x 28-32 ish -- the boxes will be marked accordingly depending on the brand).
The most reliable way to compare tire sizes is to use the ISO / ERTRO sizing. Most tires will be marked with it. Look for a number like this "32-630" (the ISO equivalent of 27 x 1 1/4). As long as you the second number, the bead seat diameter in millimeters, is 630 the tire will fit on the rim. The first number, 32 in this case, is the width of the tire, also in millimeters. That number can vary a bit without problem. If you go much wider you may have issues with clearance at the chain stays or perhaps the fork crown. Within reason, narrower won't be a problem.
27 x 1 1/4 seems to be the most common 27 inch size still made, but there are some others. Take a look at the 27 inch listings at Bike Tires Direct for example.