I can think of two things you should probably check. The first is the flasher relay, sorry I'm not sure where it is located for your vehicle (the security system probably sends its own brief 'pulse' to activate the lamps during security system arming, while the turn signals and emergency flashers require a flasher relay for prolonged on-off regulation of the lamps). The other might be a faulty dash wire (The security system signal probably gets sent directly through an electronic module, bypassing the manual circuitry of the interior turn signal lever and emergency switch). There are several manuals published for the Sienna that have the fuse/relay locations and wiring diagrams.
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