What attachment would you use ?

[devanrushton]

I've never seen or heard of these sweet gum balls before, if they're not heavy like say a green horse chestnut but more like the weight of an acorn, would a tow-behind sweeper work?

You would most likely crush some into the ground, but the majority would be swept up.

Products > 44" Lawn Sweeper

 

[bja105]

Someone else probably tried it, but what about a lawn mower with a bagger? Mine picks up my leaves, some small sticks, and some of the pine cones. You can get a brand new push mower for under $200.