Is there any other size tires that will fit on a 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).