No, you don't need one, so there. First of all, the zapper *is* a frequency generator. Another one is not the right piece of gear for testing it.
An oscilloscope is a piece of test equipment which lets you "see" volts, a very handy thing. It is the best way to determine if your creation is alive, but not the only way. Used ones are in the $100-$200 range, usually without the operating manual. Way overkill for one-time use by a novice.
Another alternative is a basic test meter of some kind: Volt-Ohm-Milliammeter (VOM), Digital Multi-Meter (DMM), or a DMM with a frequency counter. With these you can measure the output voltage of the fruit of your labor, and with the last option you can measure the output frequency. These can tell you if it alive, but not help much if it isn't. $20-$200.
Let's say you fire up your grand construction, the red light comes on, and no smoke comes flaming up. That's good. Next, bring in an AM radio. A properly working zapper creates electromagnetic energy at frequencies which extend all the way up to the AM broadcast band. Tune the radio to a dead space on the dial near the low end (540KHz-600KHz), and turn the volume up until you hear the static clearly. As you turn the zapper on and off you should hear a change in the noise. Also, with the zapper on, the noise should change as the copper handholds are touched together, kinda like Morse Code. You might have to fish around a little to find a spot on the dial which corresponds to one of the harmonics.
BTW, I *strongly* recommend **not** doing the shoebox thing. There are several websites with detailed instructions and photos showing how to build a zapper using a proto-board (prototyping board, sometimes called a solderless breadboard). No soldering, easy to verify correct connections, easy to change mistakes. There are messages in the Zapper Support Forum
and archive with links.