What are the coils of the relay rated at? 5V, OK, but what current?
I just grabbed an image from the internet to illustrate how the diode is connected. Whether or not the 555 can drive two relays without a driver transistor depends on three things: A robust battery of the -correct- voltage, how much current the relays require, and which version of the 555 is this? The TLC555 is CMOS (some others are, too) and won't source or sink enough current for any relay. The venerable TTL version of the 555 can source and sink 100mA or more. But there are probably 10 different versions of the 555 timer, all with slightly different characteristics.
Have you tried measuring battery voltage while this is operating? I would normally expect a stiff 9V supply to burn up a 5V relay coil.
Pin 4 -must- be connected to Vcc. It might work for a short time without doing so, but it will fail. That is the Enable pin.
Much clearer pictorial.