The 96 uses the last of the classic Small Block Chevies. The 99 uses the LSx based Chevy V8 and all it shares is the bore diameter and spacing. The crankshaft spigot (where you bolt the flywheel) is different, though. You will need a spacer (GM part 12563532) and six bolts (12563533), plus the flexplate from the LSx. That's to connect it mechanically. To connect it electrically may be a different matter - depending on what transfer case and case control option the 99 Tahoe has as well as what transmission it came with, it may not be able to control the transmission and t-case at all, or you may have to construct adapter harnesses.
Edit: Forgot - you will slightly have to elongate the holes for the torque converter on the flexplate or get a conversion LSx->LT flexplate. The LSx engines used a torque converter with a slightly different bolt spacing and the bolt holes are smaller and metric, whereas the LT engine was SAE.
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That said, it's not your best option, really. Were I you, I'd sell the drivetrain I've got now on Craigslist or local forums, take the money from that and get the original transmission properly rebuilt on a bench. At this point, those transmissions' seals are knocking on the door of 20 years old and things will start failing, usually right after you finished installing it! You can get most of the 4 speed GM variants built cheap if you carry the transmission in yourself.
Example:
http://dallas.craigslist.org/dal/pts/5377575867.html