If it is meant to use 4 x alkaline batteries and that is how you arrived at 6V, odds are very good that it would work at 5V.
However, there may be something you can do, depending on the design of the PSU. If it's a switcher and has just a fast silicon diode after the transformer, you could swap in a schottky and gain maybe 0.3V, which isn't a lot, but if your cam is borderline with 5.0V, surely 5.3V is close enough to 6V. If it it would have had terrible battery life, if only able to drain each cell down to 1.33V. Then again if you are using a long wire run to power it, you might have some voltage drop from that distance unless a suitably low wire gauge is used.
However, since as I suggested above, I suspect this is set up to run from a somewhat variable power input as the batteries drain, I would do something cruder if you don't have a voltage regulator. Instead I'd use the 9V PSU but put 4x silicon diodes in series on the output which will drop about 2.5V, leaving you with 6.5V, and a "little' margin for voltage drop from the wire.
However I would test this first. I mean that diode Vf drop decreases with a decrease in current so if this cam is idling at very low power draw, it could cause the resultant voltage to rise. Certainly a voltage regulator is the safer choice if you don't want to do (more, since you must still confirm proper operation) testing, but on the other hand, mains AC power is inexpensive and this must not consume much, for the lower power state you could use a resistor between positive and ground to pull the float voltage down some, or a zener diode to clamp it. Then again, IC linear or buck voltage regulators are not hard to find, nor expensive.