There really should be an online tool for doing this problem with the best available estimates, but I'm not aware of any.

Super rough idea:

There is an estimate of the difference in carbon emission per passenger km between (european) automobile and bicycle here: http://bikeportland.org/2011/12/12/new-study-compares-bicyclings-co2-emissions-to-other-modes-63536

Most bike trips will be approximately 10 km or less (is that right?). The average bike trip in the US was 3.9 miles in the summer months in 2002. Maybe longer now with better infrastructure. Google this for Canada? Really you'd need to find an average trip length for the whole year to make the calculation simpler.

http://www.bicyclinginfo.org/facts/statistics.cfm

You would need to estimate the number of one-person trips of the approximate maximum bikeable length or less that a typical person might substitute bike for car as the mode in, say, a year with their new bike. Key here is typical or average. Not everyone who acquires a bike is going to make all bikeable trips by bike. (You might be able to get an estimate of bicycle owners' typical number of bicycle trips per week by extrapolating from the Decima study of Toronto cited here: http://en.wikipedia.org/wiki/Cycling_in_Toronto).

Calculate the total number of person-kilometers in likely-bicycled trips per year. Then calculate the carbon emission for both car and bike. Calculate the difference. Then multiply that times the number of people affected (number of bikes you made available to people?).

That's the carbon saved.

Can someone check my thought process?

In reality, the numbers are so small that they will be within the margin of error at the population level, but it doesn't hurt to estimate. There may in fact be a real savings.