Liability, Liability, Liability. Hospitals hate liability and avoid it at all costs. I think it's wonderful that you want to start a program, but I will warn you in advance that you need to be prepared to fight an uphill battle.
I am a Child Life Specialist (and CPST) employed at a peds hospital here in MD and have been a tech for almost 2 years. For the past 14 months I have been working on writing policies, developing programming, and seeking funding to start some sort of car seat safety 'program' at our hospital (currently parents just watch a video). It has been no easy task and the only way I have been able to do it is by having affiliations within the hospital so that I'm able to tug the strings I need. There are so many considerations when it comes to hospitals and the 3 big ones are HIPPA laws, liability, and insurance/$.
Also, congratulations on wanting to become a CPST! That's great! Once you become certified, my best advice to you is to get out into the field (under the supervision of a senior checker) and go check seats at seat check events! You certainly will learn a lot at the CPST class, but until you can practice it and know it blindfolded (well almost), I don't recommend offering your services to a large organization such as a hospital without the guidance of a senior tech. What might be a good strategy for starters is to encourage your local hospital to provide information as to where they can get their child/baby's carseat checked...and then you become affiliated with that/those checkpoints.
If you were wanting to 'start a program' at your local hospital, and they are affiliated with JCHAO, the issues of standards and continuity of care among ALL patient populations are required. So, if your hospital deals with special needs (ie. preemies, orthopedic kids, etc) the car seat program would have to be offered to them as well...and special needs is a whole other 'bag of worms.'
It's a very admirable thing that you want to do and I say keep on it...but don't become disheartened in the process! I think sometime in the next 10 years hospitals will be more 'on board' with the whole car-seat thing, but in the meantime we all just need to keep advocating and researching!