LVL's and other engineered lumber is not recommended for outside application (unless designed specifically for outside use's). It would be great used under decks (which it's not in our state) again, unless it's designed for outside use. It can't handle the constant moisture - think plywood left outside, standing, it delaminates pretty quickly.

Lots of good suggestions - I'd use steel - but if wood is what you want to use, as MSB indicated with the nailing pattern, but use deck screws and glue with them also. Other trick, bend a flashing or sheetmetal channel over the top and down the sides some on what ever wood material you use. This sheds the water and keeps it from penitrating into and between the wood. If you use treated - look / ask for foundation grade and in yellow pine perferable. Much better quality of wood used for foundation grade material and more perservitive used in/on that grade of wood. Use a length of steel under the new "beam" where it contacts the wagon frame to keep the wood from crushing down around/into the wagon's frame (more reason to use steel stringer). 

Laminated beams put together as you perposed will tend to stay streighter and twist/warp less then solid beams cut from the tree. Personally, with no chart to look it up, I'd use steel first (painted), then your suggestion of 4 - 2x10's foundation grade treated screwed and glued together - painted with the flashing over the top.  Use enough cross braceing between the stringers also. From my experence - that's my opinion . . . .