As with megavitamin "therapy", hyperbaric oxygen as a "treatment" for autism can be accurately classed as too much of a good thing.
In medicine, too much of a good thing can be a very bad thing, and the research reflects this. Essential nutrients can easily become poisons if you consume too much of them, and many of today's common medical problems can be laid at the feet of too much consumption of essential macronutrients... and macronutrients are generally far better tolerated than the micronutrients which are used to excess in "megavitamin therapy".
In fact, hyperbaric oxygen chambers allow for a greater exposure to oxygen than would normally be theoretically possible. Put into more technical language, a hyperbaric chamber allows for partial pressures of oxygen within the lung that wouldn't be possible without active pressurization.
While this isn't the only mechanism of action for HBOT (for instance, the increased overall pressure has therapeutic value when treating decompression sickness), it's the main one, and the one which most of the various explanations of HBOT as an autism treatment hinge upon. Specifically, increased partial pressure is increased effective in-lung exposure to oxygen, and this is supposed to increase oxygen content in the bloodstream, increase oxygen flow to the brain, etc., etc.
Oxygen qualifies as a drug when used this way. As with vitamins, it's quite possible to overdose on it. This is known by a variety of names, but the most common of these are "oxygen toxicity" (for the phenomenon) and "oxygen poisoning" (for the medical condition).
Frankly, oxygen poisoning can be nasty. At lower levels, oxygen is a very good thing. At higher levels... it isn't. In fact, oxygen has neurotoxic properties at high enough doses, and these doses are acheivable even without a hyperbaric chamber.
So, if that's the case, why is it that that we don't see autistic children vomitting, suffering from tunnel vision and shortness of breath, convulsing, twitching, and generally showing the symptoms of oxygen poisoning after HBOT? The answer is actually rather simple -- and says a lot about the people peddling HBOT for autism.
The partial pressure of a gas within a mixture is equal to the percentage of the mix formed of that gas times the total pressure of the gas. So, for instance, the partial pressure of oxygen in a gas mixture which contains 20% oxygen and is held at 1 bar (normal atmospheric pressure) is .2 bar. Doubling either the percentage of gas or the overall pressure of the mix will also double the partial pressure of oxygen, yielding a partial pressure of .4 bar.
It's worth noting that a partial pressure of up to about one bar (depending on altitude, weather patterns, and just about anything else that effects atmospheric pressure) is achievable without pressurization -- 1 bar is the partial pressure of oxygen in a pure oxygen "mix" at normal atmospheric pressure. Keep that in mind as I continue.
A recent paper, using fairly typical practice for HBOT treatment for autism, used a 24% oxygen mix at 1.3 bar of pressure. Doing the math, this yields a partial pressure of about .31 bar... or roughly equivalent to what could be acheived by using a 31% oxygen mix without the hyperbaric chamber. It's actually pretty trivial to go to a SCUBA supply shop and buy a tank full of a mix that's richer in oxygen than that.
Of course, typical air is roughly 21% oxygen, meaning that the gas they used was only trivially different from room air in terms of oxygen content; the increase in partial pressure was almost entirely achieved by a raise in pressure.
While this sort of level is typical in practice, however, it's worth noting that I've seen documentation of much more "intense" regimens, ranging up to 100% oxygen at 1.5 bar. Caveat emptor -- and beware oxygen poisoning.
Finally, I suppose I should also mention that there's a second theorized action mechanism, involving gas transport via blood plasma (as opposed to the hemogloboin-based transport of red blood cells), which is increased substantially under hyperbaric conditions. This mechanism is equally implausible for reasons which are much harder to explain, but I should probably mention it in the interest of fairness.
Still -- 1.3 bar isn't enough pressure to make much of a difference there, either.