I would estimate at least one year for the following reasons:
It must be a database that, at the absolute minimum, contains Win/Loss record and Player Identifier (ID number or Email or anything else...). Most likely it would contain more than that, such as Name, and so forth. This database won't take long to create and isn't the main source of time required, except that you have to ensure that the hardware it runs on is capable of serving worldwide requests for this data, and any updates to it.
You must create an interface to this database that at bare minimum, allows searching of records, adding/retrieving data (like W/L Record, and new Players), and can handle internet/service outages. Most likely this would take the form of tournament software to assist in running tournaments. Last I knew, FFG's current strategy for this was an Excel sheet sent to tournament organizers, but it HAS been a very long time since I was up to date on this sort of thing.
In all likeliness, if you were to make the bare minimum, I would expect it to take 6 months between Development, Hardware Procurement, and QA Testing. If you were to add features to it, like being able to run a tournament that it gives you pairings for, I would expect it to take around a year, though potentially as quickly as 8 months.
The problem with these estimations is that they come from my background as a Software Engineer in a company that's built around delivering Software. FFG is not. They would need to either promote internal IT into Developers (possible, but unlikely) or hire a new Development arm of the company. (super difficult.) This is honestly what will take the most time. Alternately, they could contract out the work, but that also adds in the time to negotiate the contract and so forth.
Developing software at a business level to be used by customers is not a simple undertaking.