There are several software apps available for setting up a golf league scoring system available.
You can either run them from a PC, Mac, and possibly Android Tablets, phones, etc.
Also, there are many "online scoring" websites which will tailor to your needs, with how the league is setup.
Many of the features are setup for team competition which keep track of standings, scores, the weekly pairings, etc.
Web base apps are very good for the fact that everyone can review stats, standings, handicaps, scores, etc.
I would suggest taking a look at the following.
1. Smart Golf Scorecard - A web base system, You can setup your own league, it has Smartphone Apps for scoring.
3. Golf League Management Software
Another option is to use an Excel based program. There are several available or have someone create a spreadsheet to
track your standings, points etc.
Once you determine how to setup your league, (Team, Standings, Individuals results)
then you should post it and members here could help make suggestions of applications which would be best suited.
The initial setup is very time consuming for any league.
Once you have your database built, then entering weekly scores is usually becomes only a brief time factor.
If you elect to create or use an Excel option, I can answer most questions on creating your spreadsheet for your needs.
Good luck with searching for the many various possibilities.