His Web Shooters are cartridge-based, and he could have used his other web shooter, or reloaded with a new cartridge
Since Spiderman just got his introduction into the MCU movies, there is very little "MCU canon" references. Then again, Spiderman was created by Marvel, with his first comic appearance in Amazing Fantasy #15 (August 1962). From his very beginning, spider man used web shooters that were fueled by cartridges.
Also in the cartoon series, there were a lot of references to Spiderman using cartridges for his web shooters
The best, although indirect reference to mechanical web shooters (and therefore needing to be refilled) will be the scenes where the suit's AI mentions custom web 'settings', like ricochet web, or web grenade. Also, in the warehouse scene, you see Spidey hanging from a web hammock, and on his arm tucked behind his head, you see the bottom of his wrist, and a clear sight of his web shooter.
We already have a reference of the web shooters being mechancial, and that they run out of fluid, there was no actual reference of them being refilled (yet).Just wait, it's probably on the way.
EDIT: The original answer I've posted had come before Spiderman: Homecoming. So now we live in a world where it has happened; Spiderman is seen with web shooters, and reloading them, on screen. The best example is found on the ferry fight. About 30 seconds in, after Parker takes down all of the bad guys on the aft deck, the camera pans around as he gets in some Spidey dialog. There, you see him quickly reload a new cartridge in time to shoot out a web to a bad guy running through the garage deck towards him.