If you tear open a disposable camera... carefully... the batteries are not to be messed with or opened... you might find usable "goodies". But protect your Arduino from the ravages of ignorance by "connecting" to the flash through an opto-isolator. They are cheap, easily replaced... if you put it in a socket. Sockets are WELL worth the extra "hassle", expense. Soldering straight to pins is a newbie false economy... but has anyone not gone through learning that the hard way?
But... having said all that... if you just want the project working, the suggestion that you buy an extenal flash unit for a camera makes very good sense. It is MADE to be triggered by a switch closure, and where the switch goes is easy to find.
Above: "Buy it now" (no auction to wait for/ fight) About $25
In the other hand, maybe the sort of strobe sold for burglar alarm systems would suit you? I have one that if I supply 12v, it starts flashing about once per second. (Normal use: To draw attention to premises where alarm has been triggered.)