While I think the system isn't good as it is anyway, if we need to keep it, it should include an UI similar to what Medics have. When in need (or when you request it), icons should highlight nearby support players with ammo packs (similar to how obvious an injured ally is to a medic).
Then you also should have the option to request ammo, similar to how you can tell someone that you're coming to help them as medic.
This would be vastly more intuitive, actually enable people to potentially obtain those boxes from supports and also bypass the need for voice chat as well as any language barriers that may arise.