There's certainly no requirement for your endorser to have known you at least three years. (Three years would be rather a strange number given the experience requirement for the CISSP is five years - four with a one-year waiver.)
Using your current supervisor as an example... Even though he's only known you for one year, he would be well placed to provide an endorsement as presumably he knows your CV/resume and either he or your HR department verified your previous experience already.
When you provide an endorsement for someone you're effectively vouching for them. If your endorser has confidence you have the experience even without validating in detail all five years he is free to do that.
If he does want or need to verify your previous experience then I've always found contacting people by email is the best option as it naturally produces written evidence, but sometimes I did need to call and speak with people - afterwards I would ask them to confirm what we discussed by email.
In terms of the process, you submit an online application using the first link above. You'll be prompted for your endorser's last name and member ID. Once you've filled in the requisite details, your endorser will be sent an email asking them to provide their endorsement. At this point, if they are confident you meet the requirements they can just tick the box and provide their endorsement. If they want to check your experience they can use the details provided in your application to do that.
Once your endorsement application is fully submitted, then you play the waiting game!