The Y and C terminals probably supply 24vac when cooling is selected.
http://www.hvacmechanic.com/tstatwiring.htm
With 40VA 'formers you have 1.7A available, some portion of which is used the existing equipment. Let's say all we want to draw is 1.7/10 = 170 mA for the relay coil.
Hosfelt 45-640 has a coil resistance (or impedance?) of 160 ohms, so it would draw 24/160 = 150 mA or less. Sounds like that would work; otherwise you could look at All Electronics or Digi-key or Mouser or Jameco for a higher impedance coil. Some of their websites are a bear to search, so you might want to call.
You know, some guy back in the 70s did this and he got pretty good cooling. He used a car radiator from a junk yard. HVAC people came by to look at it because they didn't believe it.
While I have your attention, could I get from you your house sq. footage including basement and your heating gas/oil therms for, let's say, last December? I'm doing statistics on heat loss and I'm short about 20 datapoints.
Here's that stuff for my old house. Your new, low infiltration, well-insulated, house should do much better.
My house, 838 therms of NG in 90 days = 9.3 therms/day
3100 sq. ft. including basement gives 9.3/3100 = 300 BTU/sq.ft.
2658 HDD/90 = 30 HDD in one day
300/30 = 10 BTU/day/sq.ft./HDD
Multiply by 0.8 to allow for efficiency of gas/oil = 8 BTU/sq. ft./HDD
Say hi to your EE friend. I don't recommend it as a career field!