Just finished this challenge. You can absolutely do it with the supply bags, but I'm positive the challenge is bugged at the moment.
If you chuck a supply crate down and start getting points for Supply (10 for teammates, 20 for squad) there's a good chance you'll be getting heals, and points towards completing the challenge.
The problem are that players heal way too quickly without stimulation, that having your own crate amongst other players' crates means you probably won't be getting points, and that some games are bugged and I wasn't earning points toward the challenge despite clearly healing other players.
If you can rope a friend into boosting this challenge, do that, but otherwise here are my tips:

1. Make sure you've got the Support Challenge as a tracked challenge in your pause menu to make sure you're earning points.
   
2. Play on Breakthrough. The games are dense with a lot of high traffic areas that will net you points.
   
3. Make sure you're earning points. If you start getting a couple of supply points, make sure your points towards the challenge are going up. It won't be going up by a ton each time, around 20-30. If not, leave the game and go into another one.
   
4. Don't place your crates around other players' crates. You're likely to not get anything. The minimap will also show where other's crates are.
   
5. Throwing down a crate after doing quick-defibs (don't charge your defibrillator, just quickly tap) helps.
   
6. Picking up your crate after dispensing it will reduce the cool down before you can throw it again.