The equivalent of a capacitor in terms of pneumatics would be a chamber such as a metal sphere the size of a tennis ball. It would be necessary to have an external vacuum pump to evacuate the air prior to the quick release.
Auto tire houses use the reverse concept to seat a new tire on the rim. A cylindrical tank is loaded to substantial pressure by a compressor. The nozzle of the tank is shaped to fit the edge of the wheel rim and actuated by a ball valve, releasing a huge volume of air into the tire, assisting the seating process.
For your concept, you'd want to have a similar ability of opening the vacuum chamber, which may be more difficult than creating or finding a suitable chamber. For the application you describe, you can get away with a cylinder of some wall thickness but some experimentation may be required to determine the overall volume.
I'd expect that you'd have a permanently (in-use) attached vacuum pump at one end, the nozzle to apply to the solder to be removed at the other end and the release mechanism in the middle. After each use, the pump would have to remove the air, creating a bit of delay.
You may discover that a similar product using a vacuum pump exists:
It may have a chamber, or a pump with sufficient volume to accomplish the task.