The Estonian rebane goes back to the Proto-Finno-Ugric (PFU) *repä, and is cognate with Finnish, Karelian and Votic repo, Veps reboi, Northern Sámi rieban and Võro repän/rebo, amongst others. This is apparently a loan from Indo-European, as seen by the Proto-Germanic *rebaz, Persian rubah and modern Scandinavian räv/ræv.
Students of Finnish will know that the modern word for "fox" in Finnish is kettu, while repo and repolainen are archaic and/or poetic. Wiktionary states that the word kettu is "derived from the archaic noun kesi ('skin'), probably because of the importance of the fox as a skin and fur animal".
It seems that the English fox and German Fuchs have a different origin to the words in Scandinavian and Baltic-Finnic. The former words have their origin in the Proto-Germanic *fuh-, which corresponds to PIE *puk- "tail" (cf. Sanskrit puccha- "tail").
The latter words should be compared to the Spanish/Portuguese word rabo "tail", giving the modern word for "fox" of raposa in Galician and Portuguese. This word can be traced back to the Latin rāpum "turnip". The modern Spanish word for "fox", zorro, is a loan from the Basque azeri.
The journey from "tail" to "fox" is also seen in the Lithuanian uodegis "fox", from uodega "tail". Compare English bushy tail to Welsh lwynog "fox" (from llwyn "bush"; cf. Estonian vihtsaba below).
In Irish there are two terms for fox: sionnach and madra rua. The latter literally translates as "red-haired dog" (madra "dog", rua "red of hair"). Curiously the Irish term for "thresher shark" is sionnach mara "fox of the sea". The word sionnach is also to be found in Gaelic, and gives the phrase sionnach ann an craiceann na caorach "a wolf in sheep's clothing" (craiceann "skin, hide", sionnach "fox"). Curiously the Gaelic term sionnachair can mean either "fox hunter" or "will-o'-the-wisp". Sionnach in Gaelic can also mean "valve, bellows", giving the Gaelic pìob-shionnaich "uilleann pipes".
Returning to Estonia, Estonian and Võro have the word reinuvader / reinovatõr' "Rein's godparent". This comes from a folktale (Reinowadder Rebbane as told by Friedrich Reinhold Kreutzwald) where a fox becomes the godparent to an orphan.
Võro also has the term sorrõhand, which translates as "fluent tail" (sorav saba in Estonian). Learners of Finnish will in the Võro word hand recognise the Finnish häntä "tail". An Estonian term for "fox" that uses the word saba is vihtsaba "bunch of twigs tail" (cf. English bushy tail and Welsh lwynog)
See also this page for other names for "fox" in Estonian, and this page for the terms for other forest creatures. For example, echoing the Irish madra rua above, in Estonian the fox is also known as laanekoer "dog of the greenwood" (< laas "greenwood", also lehtmets) or metsapune "forest red".