Yep, File>Export>Macromedia Flash (SWF)… but…
The file will probabpy be much bigger than if you made it in Flash as AE tends to Rasterize Effects, etc.
AI files, shapes, masks are made into nevtors (and Text methinks), also for Stroke (Rasterized) use Audio Waveform or Audio Spectrum as they are treated as Vectors (mostly, Softness is dumped), look in the Help fles for more details. And an interesting quirk, when I made Swf’s from AE (back in Flash 7 times) is that if you import the swf into flash, then Export if from Flash, it is optimized and comes out smaller (quite a it sometimes) than when it went in.
And if it a AE real Video (lots of Effects that would be Rasterized and therefore HUGE), export as an FLV, Flash’s video, ask your client, surely they can handle that, unless they want to keep their product viewable with the older versions of the Falsh Player (before FLV, before 8-Ball). Ask about the FLV, if they can use it you won’t be as limited as with SWF.
My two cents.