I looking at using a linear solenoid to open and close a port at a rate of 10-20Hz with a 50% duty cycle. Will a battery powered solenoid be able to operate at this frequency?
Yes. As long as you provide enough voltage and amperage from the battery. And the appropriate coil and electronics. Batteries are no different than rectified mains.
You need to tell a lot more if the question should be answered more in detail.
Fast generally means a lot of voltage to overcome coil inductance. It is not unusual to drive a 6V solenoid with 50V. Consider using a PWM solenoid driver chip which can limit current. National semi had some. Monitoring current with a scope can tell you how fast your mechanical system is. You will see a knee when the solenoid closes and the inductance changes. You may be sad to find that the limiting factor are the mechanicals and not the electronics.