Mas Amedda probably could in the event of Palpatine's death, execute Order 66. The likelihood that he would do so, however, is very low. Amedda is a toadying lickspittle no one likes with no personal power base, who is kept around by Palpatine due to his mastery of procedure and bureaucracy. This is, after all, the same man who surrenders the Empire post-Jakku. If Palpatine has an 'in case of my death' contingency, it's far more likely he trusts it to Sly Moore instead.

Order 66 (and the other contingency orders, which yes, were ported back into Disney canon) presumably unfolds through some really complex communications protocols. Palaptine probably has to sit at his desk and input about ten different passwords and authentications to make sure it goes out properly. The actual vocal command, which can be easily faked by basically any protocol droid, is the least important part. This might make it very difficult to issue the order, assuming the Jedi take control of the Chancellor's Office and shut down his comms.

If Order 66 is not issued, everything depends on what the Clones choose. In the wake of Palpatine's death, Mas Amedda will call a special session of the Senate and a new Chancellor will be chosen. Palpatine's faction will still have the votes, so the new Chancellor will be someone they would support, which is most likely to be Tarkin. If Tarkin becomes Chancellor and the clones remain loyal to the government, they the Jedi are getting booted off Coruscant and sent into exile, and Tarkin rules as Chancellor for Life as a one-party-state style autocrat. On the other hand, if the clones pick the Jedi, then the Jedi coup succeeds, the Senate is purged, and Bail Organa becomes Chancellor a year later. Yoda and the Jedi Council believe the clones will pick them. That strikes me as a very high-risk bet, but they are guided by the Force, so maybe it isn't.

I have to say, I find the concept of Tarkin's Republic a very intriguing AU scenario.