Could the Jews begin sacrificing in the temple before the tribulation actually starts?
Yes. Nothing in Scripture precludes that from happening. In Daniel, chapter 8, we find that the Jews' daily sacrifice will be taken away, so, from that, we can infer that it will have begun again. Basically, the Jews could build their temple at any time and re-institute their sacrifices any time between now and the rise of the Antichrist.
It makes a little more sense to me that they would re-institute full temple practices before the tribulation period begins. It seems like there would be a difficult timing issue if they had to rush it right at the beginning of the Tribulation.