Does a Third Temple have to be built for sacrifices to resume in Jerusalem or can sacrifices resume without a Third Temple ? If not, what’s keeping sacrifices from resuming today, other than the requirement they be done at some location on the Temple Mount?
Sacrifices can be offered even without a Temple, but technically, the mosque on the Temple Mount is “in the way”. It's also obviously more complicated than just that technicality. Big changes, as well as all national and universal mitzvot like building the Third Temple, must be done slowly and surely, and with the initiative and support of the entire nation of Israel, and if possible, all of mankind, “for My House will be a House of prayer for all the Nations” (Isaiah 56, 7). We’re undoubtedly not there yet. In addition, Rav A. I. Kook, the first Chief Rabbi of Israel, deals extensively with the possibility (!) that there may not be animal sacrifices in the future Temple, perhaps just vegetarian offerings (see Malachi 3, 4, where the future envisages “m’nachot” from oil, wheat, wine, spices, etc., Kvatzim miKtav Yad Kodsho II, pp. 15-17). In addition, the fact that God who runs history has apparently decided that the time isn’t right yet, because we can just imagine what kind of international crisis with the Muslims would be caused by resuming sacrifices and rebuilding the Temple (this is also an halachic factor). Also, only after the blooming and settling of the previously deserted Land of Israel, the gathering of the Jewish exiles, and the return of the Sanhedrin (Supreme Torah Court)- only afterwards will the Temple be rebuilt (Megilla 17a). We should preferably concentrate first on completing those 3 previous stages, which thank God, are well under way!