What easily available chemicals do I use to turn oak light grey?
Graying is symptomatic of bleaching (oxidation) caused by air and sunshine. You can use chemical bleach for a similar effect.
See for instance: https://www.wikihow.com/Bleach-Wood.
There are a lot of guides online on the subject. This one seems pretty useful, discussing different bleaches, their safe handling, and treatment steps. Even the US Department of Agriculture has a guide on the subject.
The appropriate bleach for your application appears to be the &quot;two-part (A/B) wood bleach&quot; in which a mix of peroxide and sodium hydroxide reacts with the wood. Commercial kits are available. The USDA guide has the following to say:

One of the most powerful, satisfactory, and widely used bleaches in recent years is based on  a  strong  solution  of  hydrogen  peroxide  (30  or  35  percent  strength).  The  wood  is  first coated evenly with a solution of sodium hydroxide (caustic soda, or ordinary lye) containing pounds of the solid per 10 gallons of solution.  After this has dried for about 30 minutes, an even  coat  of  the  hydrogen  peroxide  is  applied.  The  sodium  hydroxide  applied  as  the  first coat  renders  the  hydrogen  peroxide  unstable,  whereupon  it  decomposes  and  liberates  a  large amount of oxygen in a chemically active condition, which does the bleaching. The liquid on the  surface  should  be  allowed  sufficient  time  to  dry,  during  which  time  the  bleaching  occurs; then  the  wood  should  be  rinsed  thoroughly  with  cold  water.  To  neutralize  any  remaining traces  of  sodium  hydroxide,  the  surface  should  then  be  washed  or  sprayed  with  a  solution containing 1 pound of acetic acid (or 1 pound of oxalic acid) per 10 gallons of water, allowed to stand for 15 minutes, then rinsed thoroughly with water and allowed to dry. After the wood is sanded lightly to remove raised grain the new finish may be applied.

See also the section labeled &quot;precautions&quot;.
Regarding the chemistry of the bleaching process, Wikipedia provides the following summary:

Hydrogen peroxide
Using hydrogen peroxide to delignify chemical pulp requires more vigorous conditions than for brightening mechanical pulp. Both pH and temperature are higher when treating chemical pulp. The chemistry is very similar to that involved in oxygen delignification, in terms of the radical species involved and the products produced.
Oxygen
Oxygen exists as a ground state triplet state, which is relatively unreactive and needs free radicals or very electron-rich substrates such as deprotonated lignin phenolic groups. The production of these phenoxide groups requires that delignification with oxygen be carried out under very basic conditions (pH &gt;12). The reactions involved are primarily single electron (radical) reactions.[14][15] Oxygen opens rings and cleaves sidechains giving a complex mixture of small oxygenated molecules. [...] While the radical reactions are largely responsible for delignification, they are detrimental to cellulose. Oxygen-based radicals, especially hydroxyl radicals, HO•, can oxidize hydroxyl groups in the cellulose chains to ketones, and under the strongly basic conditions used in oxygen delignification, these compounds undergo reverse aldol reactions leading to cleavage of cellulose chains.

Lignin is a polyphenolic compound that lends wood much of its color, so it is the main target of bleaching. The cellulose is best left alone, so a relatively mild bleaching protocol is best for wood.
Another important factor affecting the aged appearance, evident in the posted image, is more difficult to replicate with a chemical treatment. That is water loss, which is responsible for cracking. A quick surface treatment might not provide the highly aged appearance in the picture$^\dagger$.
Update (and somewhat off-topic)
$\dagger$ Oak is apparently among the most difficult woods to dry without it developing cracks. If the wood is already dry then it is unlikely to crack, but if it is fresh and is allowed to dry quickly (for instance by setting in the sun for a few weeks) than it is almost certain to crack.