Weapons can be upgraded in a stage if you find a specific power-up. Although you cannot carry the upgrade from one stage to another, it will benefit your weapons within that stage.

Mission 1 - Weapon Upgrade
Parachute onto the western cliff and look for the upgrade adjacent to the boxes, near the searchlight.

Mission 2 – Weapon Upgrade
Look for this upgrade on the counter of a small building at the entrance of the courtyard.

Mission 6 – Weapon Upgrade
Parachute into the western tower (the one emitting green gas) and look for the upgrade on the top floor.

Mission 10 – Weapon Upgrade
Inside the tunnel after the first 88, the upgrade can be found behind a box at the end of the first corridor.