Rashi comments on textual problems, he never gives general information that an astute reader could otherwise independently infer.
The textual problem in Judges 12:7 is the place name of Jephtah's burial. The problem is that the place name ערי גלעד "Arei Gilaad" is not otherwise known to us from the OT, and that the form of the name looks like a plural in the construct state (סמיכות), "cities of Gilaad". How then could one person be buried in several places?
Note that the problem hinges on a single Hebrew letter, a yod, the smallest Hebrew letter in the post-exilic or Assyrian Hebrew alphabet used in OT manuscripts, at the end of ער (city).
There are three ways to solve the problem:
- Assume that ערי גלעד "cities of Gilaad" is a proper place name
- Assume that the Masoratic text has a copyist mistake
- Cite extra-OT sources such as the Midrash that deal with the problem
Since Rashi is (usually) a compiler of precedent, and there was no prior opinion that "Cities of the Gilaad" is itself a proper name of a particular place, Rashi rejects the first way.
The second way is not open to Rashi for obvious theological reasons, so Rashi is left with the third way.
Remember that Rashi only cites Midrashic sources (the "drash" level of interpretation) when other explanations of the simple meaning (the "pshat" level of interpretation) are not fully satisfactory, or are simply missing, as in this case. The citing of the Midrash without other explanations could be an implicit admission by Rashi that he is not completely satisfied with this answer.
The interpretation of "Cities of Gilaad" in the current verse as a proper name in itself was proposed in 1870 by Meir Friedman in his commentary on the Mechilta, "Meir Ayin" ("Illuminate the Eye[s]"). He bases this interpretation on the similarity to other names in the Gilaad such as "Ar" (e.g. Numbers 21:28 et al.) and "Aroer".
The LXX apparently has "his city Gilaad" rather than "cities of Gilaad". This would indicate that the original text might have had a vav instead of a yod. This is one of the most common and likeliest transmission errors, as the two letters are difficult to distinguish unless the handwriting is good and the manuscript is well preserved, the vav being actually only a slightly elongated yod. This vav-yod error is commonly recorded in the Masoretic notes and is called a "kree-u-ktiv", that is, a word that is written one way but read the other way. In the case of Judges 12:7 there is no Masoretic "kree-u-ktiv" note, so we are left wondering. We must be careful though, as the LXX base might have had a yod and the reading "his city" is just the LXX's way of explaining the problem. Unfortunately there is no Qumran text to compare. teku
To answer the OP more explicitly, there is no internal OT support for the midrash (Breishit [Genesis] Rabba section 60 para 3) that Rashi cites regarding the dismemberment of Jephtah. Outside information and traditions are required to reach Rashi's conclusion.
The midrash states "Jephtah died limb by limb. Everywhere he went, an arm or a leg would fall off and he would bury it." There is further background in the Tosefta (appendix of Mishnaic material) to tractate Ohalot (tents or awnings) page 15 side A, and in tractate Ketubot (marriage contracts) page 2 side B that state that near the cities of the Gilaad there are mounds of earth that are suspect as having impurity from corpses because it was told that these mounds were used by women to bury stillborn babies and by lepers to bury limbs. The lepers would cut off a dying limb and bury it so that they could purify themselves to take part in the annual Passover sacrifice (Mishnah tractate Kritot chapter 3, mishnah 8).