I used to regularly order my own parts when it was cheaper to get it from rockauto. I would ship it straight to the garage. Are there really garages that stop you from doing that?

In case this is helpful for anyone: If you're willing to do some of the work yourself a place like Bike Sauce or Bike Pirates would work (non profit and run by volunteers). They won't just straight up do it for you, but they'll walk you through it/assist.

Edit: it's been helpfully pointed out you meant car mechanic. But I'll leave this up partially because it's funny, and partially in case it's ever of use to anyone anyway.

I think that most of them do? I recall asking a branch of Active Green a year ago and they said they can do it, but they can't give their usual garage warranty with it (which is fair).

After that, I called around to find an acceptable hourly rate (but how much time something takes is also dependent on the shop and staff).

Canadian tire will allow external parts. But, I would not do this because non OEM parts quality has degraded to the point there is a 1/3-1/4 chance the new parts won't work. The situation has gotten so bad shops are losing money replacing he same part 3X with labor.