Can I put 700c tires on rims made for 27 x 1.25"? It's getting hard to find any selection for my old Avanti 12 speed. it currently uses 27 x 1.25 inch tires. Can I put 700c tires without needing to buy new rims?
No!!!

I tried this when I was knee-high to a grasshopper and it so-did-not-work! I actually managed to get a Michelin Select 700 x 28c tyre on the 27" wheel but the inner-tube was so pinched and pierced that no amount of that gunk you can spray into tubes could save it. I did get the size wrong but after many hours of struggle 'hoped' to get a low-profile tyre on my front wheel.

The bead diameter of the 27 inch wheel is 630 mm, the bead diameter of a 700c is 622 mm (I properly learned those measurements that day). 8mm is a big difference, believe me!

If your LBS does stock 27" tyres, do not part with your cash until you have read 630 on the sidewall. If they do not stock them, ask them to get them in. This will save you the postage of mail order. Their suppliers will carry them and they should be able to get them in for next week if you ask nicely and put down a deposit.

There are specifications for wheel sizes and tyre sizes - ETRTO. Schwalbe have a good intro and list of sizes:

http://www.schwalbetires.com/tech_info/size_markings

The 27" size for your bike will be 630, 609 is the size for older bikes with 'rod' brakes and no sidewalls to the rim.

Normal 700c inner tubes will go in a 27" wheel with no problems.