May one begin to eat Chametz after Pesach prior to the conclusion of the sale?
Yes. Doing so does not border on stealing from the gentile or any other transgression. This especially applies in those sale contracts that make an explicit stipulation with the gentile that one may eat the Chametz prior to the conclusion of the return sale. [Practically so is explicitly written in the sales contracts of the Eida Hachareidis and of Rav Landa of Bnei Brak. Others however do not explicitly write this stipulation.] Even by those sale contracts that do not explicitly make this stipulation, one may be lenient to allow one to eat the Chametz prior to the end of the sale if one knows for certain that the gentile is not particular on this matter. Practically, it is proper for every Rav who is arranging a sale contract to enter this stipulation explicitly in the contract and hence merit the public.
 Rav Avraham Chaim Naah [brought in Yagdil Torah Yerushalyim 6/2610]
 Grac”h Naah ibid; Yechaveh Daas 2/64; Vaad Hakashrus of Eida Hachareidis [via phone call]; Vaad Hakashrus of Rav Landau [Via correspondence with Rav Eli Landau]; Or Yisrael 447/4 in name of Gaon of Butchach allows taking the Chametz in a) a case of great need and b) one does so before two witnesses; See Choshen Mishpat 359/1-2; Admur Hilchos Gezeila Ugineiva 1-2
 Grac”h Naah ibid
The reason: One may eat the Chametz immediately after Havdala even if the Chametz was not yet bought back and doing so contains no worry of transgressing stealing from the gentile as the gentile does not care if one eats the Chametz before the return sale. However this only applies if one takes the Chametz with intent to pay the gentile back for its market value, if in truth the gentile chooses to keep the Chametz and not go through with the return sale. Obviously if one would not agree to these terms then it would be considered stealing from him, as in this case he is particular against coming to a loss. [Grac”h Naah ibid] This is based on the law regarding taking an item from a person without his permission which is dependent on whether we assess that the owner is particular or not. [See Choshen Mishpat 359/1-2; Admur Hilchos Gezeila Ugineiva 1-2] Thus in this case certainly the owner is not particular, as the whole purpose of his bought Chametz is to be sold for market value, and thus when the Jew takes it with intent to pay the gentile back he is actually meriting the gentile, and the rule is Zachin Leadam Shelo Befanav. Now, although in Choshen Mishpat ibid this allowance only applies if one “acquires the payment to the owner through another person” [Michaber 359/2; Smeh 359/8; Admur 2] nevertheless perhaps here one may be lenient as the gentile himself [through the Eruv Kablan] himself must pay the Jew for the Chametz that he purchased, and hence in conclusion when the Jew takes it without permission, it evens out the debt of the gentile to the Jew. Nonetheless, the Or Yisrael ibid states that due to these issues it is only initially permitted to take the Chametz if one does so in front of two witnesses. See Grac”h Naah ibid
 Grac”h Naah ibid; Yechaveh Daas 2/64; See Shaarey Halacha Uminhag 2/195 which also makes mention of such a stipulation for the help of selling Chametz of a Jew which is not observant and may take the Chametz on Pesach; See however “Seder Mechiras Chametz” p. 178 of Rav S.D Levin which states that this condition established by the Rebbe only applies to the sales contract of the store, and not to the general sale contract.
The benefit of this stipulation: Making the stipulation explicit in the contract avoids all worries and issues that were discussed in the previous footnote. [Grac”h Naah ibid]
 This stipulation was placed by Rav Avraham Chaim Naah, who was the secretary of the Eida Hachareidis at that time. The stipulation was brought to the attention of Rav Bengis who was the Chief Rabbi of the Eida Hachareidis at that time, and he praised the idea.
 In the words of Rav Eli Landau “Every [respectable] Rav of all generations made this stipulation both orally and in writing with the gentile, in order to allow the Jew to take the sold medicine if he so needs. The gentile gives explicit permission to the Jew to take the Chametz before it is sold back on condition to pay him for it.”
 The contract of Rav Ashkenazi of Kfar Chabad does not contain this stipulation, and neither does the contract written by Rav Levin in his Seder Mechiras Chametz.
 See Choshen Mishpat 359/1-2; Admur Hilchos Gezeila Ugineiva 1-2 and “The reason” brought in previous footnotes for an explanation of this matter. Practically, while the Grac”h Naah seems to side there is no issue with taking the Chametz even if it was not stipulated in the contract he explicitly records the Or Yisrael ibid that allows doing so only in front of two witnesses.
 Grac”h Naah ibid; Rav Eli Landau ibid