I don’t work for Lifetime Fitness but I know they have a similar service as to the one you are thinking to offer.
You might want to give them a call and find out more specifics. I hope this helps.
There should be no issues if you are not the one actually drawing blood and testing it. The Phlebotomist and the lab will have their own insurance. You will most likely just arrange the service to be provided.
If you handle any of the information involved, you will have to be very secure with it. That would be the only liability that I can see. But it is a good idea to confirm any potential issues with the people involved and an attorney.