Is the practice of ahlal badri proven from Sunnah?

In this practice, the names of the participants of the battle of Badr are recited in case of a theft or some evil deed. After the recitation, the perpetrator is befallen by some calamity ranging from physical injury to mental problems.

Such an act is a bid’ah (innovation) and is not permissible unless you can provide proof that the Prophet (ﷺ) did it, allowed it or approved of it. Any act of worship not approved by the Prophet (ﷺ) is bid’ah.

As stated in the hadeeth narrated by 'Aa'ishah (may Allaah be pleased with her), who said: "The Messenger of Allaah (ﷺ)  said: 'Whoever innovates something in this matter of ours that is not a part of it will have it rejected.'" [1]

Furthermore, the Prophet (ﷺ) stated repeatedly that: "Every newly-invented thing is a bid'ah (innovation), every bid'ah is a going astray, and every going astray will be in the Fire." [2]

The Prophet (ﷺ)  used to say, when beginning a khutbah (sermon): "… The best of speech is the Book of Allaah and the best of guidance is the guidance of Muhammad. The worst of things are those which are newly-invented, and every innovation is a going astray…" [3]

References: 

[1] Reported by al-Bukhaari, Fath al-Baari, no. 2697

[2] Reported by al-Nisaa'i in al-Sunan, Salaat al-'Eedayn, Baabkayfa al-Khutbah

[3] Reported by Muslim, no. 867