H.W.
Would this be considered bad for you, administering medicine without a current license? Are you insured to do this? What if your MIL has a bad reaction?
These are the reasons I would give her: concern for her health coming first and foremost. At a doctor's office, if she has a reaction to the injection (either the injection itself or the administration) then she's likely closer to getting the medical care she'd need.
Or you could always take it on yourself and just say "You know, the last time I gave you the shot, X happened. (However you want to explain it.) This made me realize that I'm not the best candidate for doing this, and I want to make sure you stay healthy. I'm not confident that a mistake would be a minor thing, and I cannot discern between blood and the medication, which puts me at a loss. You should plan on having your medical provider do this. I'd really hate to hurt you in any way, and a mistake could cause you serious problems."
I think you could suggest this in a concerned and caring way, not "I don't want to" but "I don't think I'm the best person to...". She'll likely be disappointed, but it could be worse!