Should helicopters fly when they the lack power for vertical climb out of ground effect?
Up to a certain speed, which will be in the POH, the greatest power requirement is when hovering out of ground effect.  Next is hovering in ground effect and power required in forward flight is the lowest, and decreases up until a point where the increase in parasitic drag starts to increase the power required.

See a typical power required curve.


From the AOPA blog

So it is possible to not have enough power margin to be able to hover OGE but enough to hover IGE.

To perform a normal take-off (ignoring a lot of side effects such as need for pedal), come to a normal hover height, about 2-3 feet, then push the cyclic forward.  Increase collective smoothly to maintain altitude and allow the aicraft to accelerate which it will, since increasing collective increases power and the rotor thrust now has a forward horizontal component. As significant translational lift kicks in at about 10 kts, a reduction in collective is required to avoid climbing early.  You don't want to be climbing with high pitch, and therefore high drag, if the engine quits.  When reaching about 45 kts or so, pull back on the cyclic to adopt a climbing attitude.  The aircraft will settle at about 65 kts.

To perform a "creeping" take-off, increase collective to become light on the skids.  Gentle forward pressure on the cyclic will start a slide on the ground. As translational lift kicks in, apply a little back pressure to lift gently to climb to about 1-2 feet to remain in maximum ground effect and continue to accelerate.  Wait until 45 kts is reached, and off you go.  During this manouver, the collective is left where it is until established in the climb, aside from small adjustments needed to keep the altitude constant during the creep.

I am not sure if it is a certification requirement, but every helicopter I've flown is capable of safely flying when there is not enough power margin to hover OGE, and most helicopters I have flown are on the weak side (R22, B206, B206L, S300, Enstrom 280) so I guess all helicopters can do so.

I checked a few types certs and I can see nothing about creeping take-offs. I am reasonably sure that the only limits that apply are for RPM, MAP, torque, N1, N2, temperatures etc. As long as you don't exceed those and are in accordance with the POH, then it doesn't matter how you get airborne. As you can see from the power required curve, the power required for forward flight is a great deal lower than for hover. 

I've copied in your comment:


  I understand that vertical climbing is not hovering, but neither is it
  "flying" in the same sense that translational flight adds lift.
  Positive vertical climb out of ground effect with no translation is, I
  deduce, the hardest thing a helicopter can do. The maximum rate of
  climb in that attitude seems like something you'd have charts for, but
  I don't even know a term or acronym for it.


Because of translational lift, a helicopter requires less power to climb when flying forward than it does to hover, in or out of ground effect.  The greatest power requirement is when hovering OGE and increasing the hover height requires more power still so yes, climbing vertically OGE requires more power than any other regime.  There is no chart, or term for this that I am aware of simply because you should not do it and the POH will detail restrictions, not least, the "dead mans" or avoid curve.



http://flyinghelicoptersinalaska.blogspot.co.uk/2010/04/why-dont-helicopters-go-straight-up-and.html

The only time you should ever climb vertically (a "towering" take-off) is when operating out of a confined area when it would not be possible to get out without clearing the trees, walls or whatever.  Some pros do it all the time, e.g. power line inspection/repairs, logging and so on but they accept a risk of operating outside of the limits and are certified and insured appropriately.

The avoid curve shows the combinations of height and speed in which test pilots (i.e. the best) have demonstrated that a safe landing is possible if the engine quits.  You should not fly in the shaded area so you can see (and that chart is similar for all helicopters) that a towering take-off should never be used unless there is no alternative, and only then, for the shortest possible time.  Coming into a high hover, say 50 feet, or continuing a vertical climb OGE is prohibited by virtue of it being in the shaded area(s) of the chart.

Helicopters are not always intuitive.  Many people are surprised by the fact that climbing needs less power than hovering, and the faster you go (up until that point where the parasitic power increases dramatically), the less power is required.  Imagine a car where less power was required to accelerate and go faster!