One of the Laws is not to sacrifice anywhere but the required place (which is the Temple in Jerusalem). By not making sacrifices, we are keeping the Law.
Actually, the required place could be the Tabernacle. Nowhere in the Five Books of Moses is the Temple even mentioned, as it did not exist when they were written. Indeed, the Samaritans, who have basically the same Torah, with a spurious interpolation regarding Mount Gerizim, continue to offer animal sacrifices.
There is nothing in the Old Testament which would preclude a kohanim from officiating over a valid animal sacrifice, which indeed, is why some Karaites are actively seeking to obtain a red heiffer so as to resume sacrifices.
The idea that animal sacrifices cannot be offered in Judaism since the destruction of the Second Temple, and the related idea that the Third Temple will not be built until the coming of the Messiah (which not all Jews agree on, hence the efforts to build it now) stem from Rabinnical Judaism; they represent the response of the Pharisees to the destruction of the Temple and the depopulation of Jerusalem by the Romans, events which facilitated a complete Pharisaical dominance over those Jews who did not convert to Christianity (a great many Jews did convert), which would remain unbroken until the emergence of the Karaite movement.
With, of course, the notable exceptions of certain geographically isolated Jewish populations, like the Beta Israel of Ethiopia and some Indian Jewish communities which later became Rabinnized (I think the evidence is clear that most of the Kochin Jews became Nasrani Christians in the 1st-3rd centuries, and the Bene Israel were on the other hand too remote and obscure to become subject to the Sephardicization which later radically reshaped Kochin Jewry).
The Beta Israel are increasingly subject to rabinnical influence, but continue to have priests, and animal sacrifices. I consider this fact by itself the most compelling refutation of the idea that the Temple is required to have sacrifices; the Beta Israel are halakhically Jewish, unlike the Samaritans, and represent a tradition of Judaism which thrived outside of the oppressive influence of Pharisaism.
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Of course, animal sacrifices or not, none of this is by itself efficacious. I pray that the Jews will be saved according to God's special grace, however, at present, animal sacrifices of the Beta Israel and the davening and torah study of the Charedim are equally lacking in grace, on their own terms. The surest way for any Jew to obtain salvation is to receive baptism and partake of the Eucharist, obtaining the fulfilment of the Torah from the hand of the Messiah, God incarnate.
