I am not sure how Flash works when embedded in a PDF i.e. if Flash Player must be installed separately, or if the Reader plugin (C:\Program Files\Adobe\Reader 8.0\Reader\plug_ins\Multimedia\MPP\Flash.mpp) is sufficient.
You mentioned "Flash Player 10", so there is the possibility that the installed Flash Player may be used in addition to the Reader plugin. In that case the question comes up: which Flash Player, the ActiveX (for Internet Explorer) or the browser plugin (for other browsers like Firefox)?
Can you experiment with that, e.g. uninstalling all Flash Players in a test environment, and see if you can still play the embedded SWFs? If not, install either ActiveX or plugin Player, and see which one is used?
FP uninstaller: http://kb2.adobe.com/cps/141/tn_14157.html
Legacy media (which is the only thing you can create if you're targeting below Acrobat 9) uses the *external* instance of whichever renderer has the registered class handler for the file extension in question. It may be Flash Player, but in some rare cases users will have another program handling SWF files. There is no embedded copy of the Flash Player runtime in Acrobat or Reader prior to 9.
Legacy media has both technical and security issues, and as such it has been depreciated in the current versions and is blocked by default. If an external player instance has been created then the multimedia trust has been granted (either by the user or because their version of Acrobat/Reader is too old to care), but the handoff between the external player and the Acrobat/Reader process can be affected by security in the Flash Player model.
In Acrobat/Reader 9 and later, all native rich media annotations (SWF, audio and video) will play through the embedded Flash Player runtime, irrespective of what else might be installed on the user's system. This ensures security is preserved, as the sandbox rules for the embedded runtime are significantly tighter than for the standalone version.
It is possible to switch a rich media annotation into a legacy media annotation in Acrobat 9, which will then revert to the external player, however the ability to create legacy media has been removed from Acrobat X.
In this case, without seeing the PDF file I can't say why it's not working.
Thanks. We have finally gotten someone in the client's IT department to
agree to take a look at this. Hopefully they will be able to shed some light on this.
As I noted in my original post, the client is currently NOT using Acrobat 9 or X, so using the new, native Flash support is unfortunatly not an option at the moment.