Answer: The Shulchan Aruch (OC 296:2) writes that one cannot say havdala over bread. One may use beer, however, providing that it is chamar medina, a national beverage. This is different to Friday night kiddush where the Shulchan Aruch (OC 272:9) writes that one should use bread for kiddush rather than other drinks, though similar to the daytime kiddush when beer would be second best.
The Rema (OC 296:2) writes that the minhag is to use beer for havdala on motzaei Pesach as one appreciates beer more then. The Taz (296:3) and Shulchan Aruch Harav (OC 296:10) note that this is subjective, and if one prefers wine then one should use wine for havdala.
R’ Chaim Volozhin (Maaseh Rav 185) notes that the Vilna Gaon was particular to eat chametz on motzaei Pesach. The Taamei Haminhagim (Kuntres Acharon 593) explains that he wanted to demonstrate that the reason that he avoided chametz for the past week was only because it was a mitzva to do so. For this reason, there were various acharonim who were particular to use beer for havdala (See Nitei Gavriel, Pesach 3:21:2). The Torah Temima (Shemos 12:168) even relates that the Vilna Gaon himself used to use beer for havdala on motzaei Pesach.
The Kaf Hachaim (OC 296:26), however, disagrees, writing that even if one prefers beer, one should use wine for kabbalistic reasons. Nonetheless, other poskim aren’t particular about this. Thus, the Aruch Hashulchan (OC 551:26) writes that it is ideal to use beer for havdala during the nine days when it is best not to have wine.
The Aruch Hashulchan (OC 448:28) stresses that the Rav should buy back the chametz that’s been sold as soon as possible as one can’t take it until then.
In conclusion, some have the minhag to use beer for havdala on motzaei Pesach. They should only do so if they like beer, and must either buy it then or wait until the chametz that they sold has been bought back.