Sometimes during battles, a character will get the 'Convoy' option, letting them access the convoy during the battle. It's super handy if I break a weapon or need a Vulnerary, but it doesn't show up as an option most of the time. What's worse is that I can't figure out when it does and doesn't show up. When does it show up?
I don't know about Awakening specifically, but according to my past experience with every Fire Emblem title where Convoys were available, the condition to access it would be by having your character land right next to your main character (in this case: Chrom) or [thanks @AdamP] when your character is paired up with him (Chrom)
The main character obviously will always have access to the convoy.
Here is how convoys work across the Series:
Fire Emblem 4: Seisen no Keifu / Genealogy of the Holy War
There is no convoy there, but items are directly sent to or can be fetched from castles.
Fire Emblem 5: Torakia 776 / Thracia 776
Similar to 4, but with shop-like places instead of castles.
Fire Emblem 6: Fuuin no Tsurugi / The Sword of Seals
Fire Emblem 7: Rekka no Ken / Binding Blade
Here, Accessing the convoy is done by having a character land beside Merlinus, a character that cannot fight and whose sole purpose is to act as the convoy.
It is worth noting, that Merlinus' death is not permanent and (in 6 only) items can only be sent to him as long as he is still on the map.
Fire Emblem 8: Seima no Kouseki / The Sacred Stones
Works similar to the convoy in 6 & 7, only this time, the convoy travels with the main character: Eirika or Ephraim.
Once they reunite, you can only access the convoy through the lord whose path you chose to follow.
Fire Emblem 9: Souen no Kiseki / Path of Radiance
Fire Emblem 10: Akatsuki no Megami / Radiant Dawn
Here, the convoy can only be accessed in between chapters.
Fire Emblem 11: Shin Ankoku Ryuu to Hikari no Tsurugi / Shadow Dragon
Fire Emblem 12: Shin Monshou no Nazo ~Hikari to Kage no Eiyuu~ / New Mystery of the Emblem ~Heroes of Light and Shadow~
Just like in 8, the convoy is accessed through the main character: Marth.
Fire Emblem 13: Kakusei / Awakening
Just like in 8, 11 and 12, the convoy is accessed through the main character: Chrom
Characters that act as the convoy can always access it, regardless of the condition, as soon as the convoy becomes available ingame, even Merlinus can.