Who has the most influence, outside the final decision maker, with regards to choosing technology vendors, data integrations, and API connectivity?

Really depends on the organization. In some organizations everything needs to be run by upper management in regards to technology. In other orgs it's the product who runs the whole show, and in other one's it's Engineering which gets to decide on what to use. Ultimately though it's the people who control the purse that will have most influence because it has to come out of some budget and that will drive your technology choice (if this shiny new tech is 2x the budget you have, you can forget about it).

Agree with Anthony. 

Also, I think the most critical person is not on the list though..... the Enterprise Architect.

Who has, or who should have?

The list is a good summary of the choke points of decision making when it comes to most of the above.

But at the end of the day the decision should be influenced by what best solves the problem, and the people who are best served to influence that are the end consumers.