Does the US postmaster have the power to decide by themselves to whom to award a $6.5 billion contract?
Probably yes, in fact for the rest of the government the person who ultimately approves a contract has to have authority to spend government money. The way the executive branch of government works is based on Article 2 of the US Constitution
The executive power shall be vested in a President of the United States of America.
Effectively, all power the executive branch has, including the authority to spend the money allocated to it by Congress, is vested in the President, who then delegates that power to other people. Depending on the power, those people are then empowered to delegate part of their authority to their subordinates, and so on. The complicating factor for this question is that USPS is not a normal executive agency, it's a public corporation, so it gains its spending power by its incorporating statute rather than as a part of the executive branch's general spending authority. In terms of how it delegates power it probably works the same way, such that the Postmaster General has the ultimate authority to spend the money USPS has.
While it looks like this wasn't a sole source (only one bidder) award, if it had been USPS appears to require that those are approved at the USPS Vice President level, but you can see that that is because of the way spending authority is delegated, so it would not apply to the Postmaster General - https://about.usps.com/manuals/spp22010/html/spp2_321.htm#ep975601
The Vice President, Supply Management, has delegated to the Portfolio managers $10 million worth of approval authority for noncompetitive contractual actions, and these managers may redelegate some of this authority to contracting officers in their organizations.
Is there a standard government bidding process for large contracts like this?
For executive agencies there are two, sealed bids and competitive proposals. In both cases a "Request for Proposal" is made available to the public, which has instructions and requirements contractors can use to draft a proposal. In a sealed bid process, those proposals are sealed when they are received until a selection date, at which point all bids are unsealed and read publicly, and the government will pick the one that is the lowest bid while meeting the requirements. In a competitive proposal process the bids are instead reviewed privately by the government and evaluated, and the government then sends the bidders reviews of their bids with explanations of where they think the bids are deficient, with an opportunity for bidders to revise and resubmit their bids. After that point the government selects a bid based on "best value," which is going to be subjective based on requirements but is effectively which one will meet the requirements plus whatever extra value is actually useful to the government, for the lowest price.
It looks like USPS uses a similar process to competitive proposals - https://about.usps.com/manuals/spp22010/html/spp2_235.htm
Once the deadline for proposal submission has lapsed, the proposal evaluation team records the proposals received and reviews each proposal to determine whether it meets the requirements that the Postal Service outlined in the RFP
The [Contracting Officer] will question the supplier in regard to a failure to acknowledge an amendment to the proposal or if the proposal contains a suspected mistake.